None of which, alas, could really be said about me—especially when I'm making dumplings at the table in the hallway outside the kitchen. Even though A-Mui has shown me how to put a dollop of seasoned pork onto a round flour wrapper and crinkle it together with one hand, I just can't get it right. My folds are off. Filling spills out. Eventually, though, I fall into a rhythm, and once I've accumulated about 50 mostly misshapen dumplings, A-Mui and my mother-in-law, a petite, elegant woman named Mei-Mei Chen, pluck out a single one and discuss it between themselves at length. Finally, my mother-in-law declares, in careful English, "A-Mui says this one is perfect."

It's a minor success, and in this household I'll take what I can get. It's been more than ten years since I met Jean Liu, the only daughter of Mei-Mei and her husband, Kan-Nan Liu, but I've never quite found my place in her family. At first, the reasons seemed obvious. I'm not Taiwanese, and I speak only a little Mandarin, the official language of Taiwan, and hardly a word of Taiwanese (the dialect of the descendants of the Chinese who arrived on this island 400 years ago). Also, I am not a doctor. Jean's mother is a general practitioner, and her father is a brain surgeon. A brain surgeon! I'm a writer, which in the Liu family cosmology ranks well below doctor, computer-chip designer (Jean's brother's job), and even artist—but above day laborer.

There has, however, always been one tiny, tenuous point of connection between Jean's parents and me: food. Or, more specifically, my fascination with Taiwanese food, which, I learned over time, is heavily influenced by Japan (the island was a Japanese colony from 1895 to 1945) and incorporates regional flavors from all over mainland China, thanks to the influx of immigrants after 1949, when Chiang Kai-shek and his Nationalists, having lost the Chinese civil war to the Communists, retreated to Taiwan, bringing millions of refugees. Despite its many influences, the cooking on this small island (about the size of Maryland) maintains a stark individuality. It is defined by its ingredients—herbs like basil, offal such as intestine, lots of seafood—and also by its structure. Big, heavy dishes are rare, and xiao chi, or small eats, are the norm, found everywhere from night markets and street corners to full-scale restaurants and homes. It's all about snacking.

On annual visits from our home in New York City, Jean and I would sit at the table with her mother and father on the far more modern second floor of their long, narrow apartment, eating watermelon and crunchy green guavas and discussing where to get the best niu rou mian, spicy beef noodle soup garnished with pickled mustard greens, or mian xian, a gelatinous wheat noodle soup. At mealtime, there were only three subjects of conversation: what we'd just eaten, what we were eating now, and what we'd eat later.

I reserved my greatest enthusiasm for A-Mui's simple, satisfying home cooking. And in A-Mui herself, I saw an opportunity. Roughly pear-shaped, with twinkly eyes and a crackly, joke-making voice, she was (like me) an outsider who (unlike me) had become part of the family. Perhaps, I thought, if I could learn to cook as she did, I could prove my merit to Jean's family. I even mentioned the idea to A-Mui on my first visit; it was politely smiled at and forgotten.

Eight years later, Jean and I are married and her parents have resigned themselves to my presence, perhaps because Jean is pregnant with our first child. A-Mui, meanwhile, is nearing 70 and considering retirement. Now is the time, I realize, to make good on that old vow; when A-Mui leaves, her knowledge will leave with her, and the recipes that sustained my wife and her family for all those years will be mere memories. So, with my wife's blessing, I've flown to Taipei by myself for a week to learn A-Mui's secrets—to preserve her recipes and thereby make myself an integral, indispensable member of the Liu family.

The building that houses the Lius' multistory apartment has a wonderfully faded grandeur, the Ionic columns on its façade wrapping elegantly around a street corner in the Ximending (West Gate) neighborhood. It was once the Capitol Hotel, and then the Capitol Hospital, and now it's been divided into various businesses, including a clinic, which Mei-Mei runs, and the Liu family's apartment. This is typical of Taipei, a city of 2.6 million that has, in the span of 60 years, speedily but haphazardly transformed itself into a hypermodern metropolis, with citywide wi-fi and the world's second-tallest skyscraper. Still, lots of the old bones remain. It's progress, tempered by practicality.

The first morning of my visit, I am on the street outside the building at 6:00 a.m., bleary from jet lag, to join A-Mui on a trip to the market. Following her through the streets at this early hour, I can barely recognize the neighborhood. At night, Ximending is a frenetic place. If you're a Taipei teenager, it's where you go to shop for trendy Japanese clothes, watch movies, and eat in goofy theme restaurants. It's noisy and neon and innocently joyful. But this early in the morning, Ximending is still and silent. In the postdawn light, other facets of the neighborhood stand out: a municipal recreation center with a Japanese cafe on the ground floor; an all-night noodle shop that's closing up; the elementary school that Jean, her mother, and her grandmother all attended. And the market.

Some Asian markets have a romance about them—a festive, rainbow spirit. Not the one in Ximending. It's in the basement of an unassuming apartment building, and the stairway entrance looks like a portal to Hell. Crimson meats, bulbous organs, and whole pigs' faces hang from hooks; monstrous slabs of deep-sea squid are rehydrating in plastic tubs; a man is using a blowtorch to singe the hair off pork trotters. The floor glistens, and the yellow incandescent lights glow like the Devil's eyeballs. My mother-in-law didn't want me to come here: Wouldn't I prefer the pretty fruit market instead?

I trail A-Mui as she browses the inventory: fish dragged hours before from the South China Sea, black-foot chickens, pungent greens trimmed from mountain slopes, more kinds of bamboo shoots than I knew existed. By the time we're back home, I'm beat. I lie down for a quick nap, and when I wake up, around 10:00 a.m, A-Mui has already cleaned and chopped everything and is starting to prepare lunch.

Working with that single wok, A-Mui methodically adds ingredients—slices of ginger to fry in oil, then chunks of kabocha squash, then water—and in minutes, when the squash has softened, has a completed dish. And then another: scallions and fat shrimp, stir-fried with a splash of rice wine and finished in a flash. Next is kong xin cai, or water spinach—boom, done. Fried spareribs and cubed daikon in broth: ta-da!

And that's just the first meal. Over the course of the week we repeat the morning's schedule again and again, using different ingredients and different dishes, with A-Mui doing her best to explain her methods in Taiwanese, which I don't understand, and me asking questions in bad Mandarin, which she doesn't understand. The dishes are outwardly simple, and so is the repertoire of cooking sauces at her disposal (dark soy, sweet soy, rice wine), but soon I pick up on some subtle techniques. To almost every dish, I notice, she adds a spoonful of tiny beige pellets—powdered mushroom broth—which adds umami depth without the ill effects of MSG. She shows me how to use the little dried fish I've seen in market stalls for years, stir-frying them with chiles and peanuts, or adding them to shan shu, a wild leafy green. A-Mui is also a master of deep-frying. She uses sweet potato flour to give a light, crunchy crust to everything from chicken wings to pork cutlets marinated in sour hong zao, a pink paste made from fermented rice. And she fries foods at two temperatures, first over a medium-high flame, to cook it through, and then cranked up high, to fry the exterior to a crackling brown.

What never ceases to amaze me as I watch her is how she blazes through the cooking. No matter what she's making—long beans with ground pork, an eggy custard studded with shrimp and mushrooms—lunch is ready in 20 minutes, and either she or her husband, A-Hang (who does odd jobs for the family and is as skinny and shy as A-Mui is big and friendly), will carry the tray of dishes downstairs to the big, round dinner table on the second floor.

The more time I spend with A-Mui, the more I want to know about her, about where she came from, where she learned to cook, but the language barrier keeps me from asking. So, one day over lunch, I ask my mother-in-law about her. In central Taiwan around the middle of the last century, Mei-Mei explains, girls were often "adopted" into the families of the men they were to marry. This was the case with young A-Mui, and she lived with A-Hang's family for years. But A-Mui wasn't happy, so she ran away to Taipei and cooked for a few different families, eventually winding up with the Lius. The employment gave her economic freedom (she not only put her kids through university but invested in land, too) and, I imagine, helped her reconcile herself to marriage with A-Hang.

Most nights during my visit, I try to get out of the house for dinner—after a full day with A-Mui and my in-laws, I've had all the family and all the cooking I can take. Many evenings I meet up with Jean's cousin Vince. About my age and very tall, he's a dentist who speaks excellent English, longs to continue his dental studies in America, and loves to eat. On our first night out, he treats me to a classic snack pairing—rou gen, a thick soup with bits of pork, and lu rou fan, luscious stewed pork over rice—at a bustling, open-air stand that, as many Taipei places do, blurs the distinction between restaurant dining and street food.

The next night, we head straight for Yong Kang Street, in a working-class neighborhood that has some of the best restaurants in the city. Here we consider stopping at James Kitchen, a Taiwanese-style izakaya, or pub, that grills mackerel as perfectly as any place in Tokyo and serves it alongside stewed pig intestine. But we bypass it for the famous soup at Yong-Kang Beef Noodle, whose namesake dish is spicy and sweet with tender braised beef and pickled mustard greens. Then we go around the corner to Ice Monster, where we split a plate of crushed ice topped with sliced mangoes and mango ice cream.

On another outing we go to the Raohe Street night market. Night markets are a Taipei institution, bustling lanes where vendors sell cheap jewelry, clothing, and, most important, food. The market has a mind-blowing variety of dishes: skewered fish cakes and grilled cuttlefish and my favorite, hu jiao bing: black pepper buns. A northern Chinese import, hu jiao bing is wheat-flour dough stuffed with peppery pork and baked in a cylindrical clay oven that calls to mind an Indian tandoor. The line for them is long—30 minutes, at least—but Vince is patient, unfazed. Everywhere we've been, he's exuded a sweetly geeky enthusiasm not only for the food but also for my friendship. He seems inordinately pleased to be hanging out, and frankly, I'm not sure how to react. Who am I to deserve such kind attention? Then I realize that Vince is the only family member of his generation still living in Taipei: his brother is in Tokyo, his cousins in America. He's an insider looking out—me in reverse—and my presence offers him a taste of another world. And so we wait for our buns, happily.

The next day is a Sunday, and about a dozen aunts, uncles, and friends are coming over for lunch. Today's meal is special not only because of the company but because A-Mui is showing me how to make two dishes traditionally served to new mothers—which is what Jean will soon be. One is a creamy stew of pigs' feet that's supposed to stimulate breast milk production. The other is ma you ji, chicken cooked in black sesame oil and rice wine and served over thin noodles. As usual, the ingredients are minimal, the flavors boundless.

Also as usual, A-Mui is a whirlwind, getting ten dishes into and out of the wok in an hour. I help her make a whole red snapper steamed with black beans, a winter melon soup with clams, and a dish called cang ying tou, or fly's head, which is a fiery pile of ground pork, garlic chives, and chiles. Everyone clusters around the table, chopsticks dart in and out of the bowls, and A-Mui observes with satisfaction. Mandarin, Taiwanese, and English bounce around, and Jean's father, who's been working long hours all week, beams a proud, beatific smile at the clan.

Amid all this action, I feel a bit lost, but contentedly so. For this family—my family—eating well is the normal course of affairs, and the greatest sign of acceptance is not to be singled out for special attention but just to be offered a bowl, a pair of chopsticks, and a seat at the table.