Instructions

1. Make the chicken: To a large pot, add the chicken, cilantro, bay leaves, celery, garlic, carrot, chile, and onion and cover with the stock. Season with salt and black pepper and bring to a boil. Turn the heat down to maintain a strong simmer and cook breast-side down for 25 minutes. Rotate the chicken breast-side up and cook until it pulls easily from the bone, about 15 minutes more. (Do not turn off the heat; continue to gently simmer.) Using tongs, transfer the chicken to a large bowl. When cool enough to handle, remove the skin and transfer it back to the stock, then pick over the carcass, shredding and reserving the meat in a large bowl. Transfer the carcass back to the stock; allow the stock to simmer while you make the sauce.
2. Make the salsa verde: In a medium pot, cover the tomatillos with water, bring to a boil, then turn the heat down and simmer until dull green and softened, about 10 minutes. Using a slotted spoon, transfer the tomatillos to a blender along with the cilantro, sugar, a pinch of kosher salt, the garlic, half of one onion, and ½ cup of the tomatillo cooking liquid. Blend until smooth.
3. To a large skillet over medium-high heat, add 1 tablespoon of vegetable oil. When it’s hot and shimmering, add the sauce and turn the heat to medium-low. Cook until thickened and reduced slightly, 10–15 minutes. Pour into a bowl and set aside. Wipe out the skillet.
4. Over a gas burner (or under a broiler preheated to high), cook the poblanos, turning occasionally, until charred on all sides, about 15 minutes total per pepper. Using tongs, transfer to a zip-top bag, seal, and set aside for 15 minutes to steam. Keeping the warm poblanos in the bag, transfer one pepper to a cutting board. Peel and discard the skin and seeds, then slice the flesh into ¼-inch strips and transfer to the empty skillet; repeat with the remaining peppers. Slice half of one onion into half-moons and transfer to the skillet along with 1 tablespoon of vegetable oil. Season with salt to taste, turn the heat to medium-high and cook, stirring occasionally, until the onions are soft and beginning to brown, 6–8 minutes. Transfer to a bowl and set aside until lukewarm, then stir in ⅓ cup of the crema. Return the skillet to the stove.
5. Through a fine-mesh strainer set over a bowl, strain the stock and discard the solids. Finely chop the remaining half an onion and transfer to the skillet along with the zucchini, yellow squash, and 1 tablespoon of vegetable oil. Season with salt to taste, turn the heat to medium-low and cook, stirring occasionally, until the vegetables are soft and beginning to brown, about 15 minutes. Transfer to the bowl with the chicken along with ⅓ cup of the crema and ⅓ cup of the stock and stir to combine. (The texture should be wet but not soupy; add more crema and stock if needed. Reserve the remaining stock for another use.)
6. Wipe out the skillet and return it to the stove. Add enough oil to just coat the bottom of the pan and turn the heat to medium-high. When it’s hot and shimmering, working in batches, fry the tortillas (flipping halfway through cooking) until crisp but not crunchy, about 1 minute total per batch. Transfer to a paper towel-lined plate.
7. Assemble the budín: Position a rack in the center of the oven and preheat to 350°F. In a small bowl, toss together the asadero and Oaxaca cheeses. In the bottom of a 9- by 13-inch baking dish, arrange one layer of tortillas side by side (they should not overlap; gaps are ok). Using a spatula, distribute about half of the chicken mixture over the tortillas in one even layer, spreading it to the edges. Ladle 1 cup of the salsa verde on top and spread to the edges, followed by one third of the remaining crema and one third of the shredded cheeses. Cover with another layer of tortillas, spread evenly with the poblano mixture, then (as in the previous layer) top with 1 cup of the salsa verde and another third of the crema and cheese. Follow with another layer of tortillas and repeat the previous layering of sauce, crema, and cheese. (Reserve the remaining sauce.)
8. Bake until crisped at the edges and the cheese has melted, 35–45 minutes. Set aside for 15 minutes to cool slightly, then serve, passing the leftover sauce and additional crema on the side.