Instructions

1. Make the filling: Position a rack in the center of the oven and preheat to 400°F. Using a sharp knife, remove the pheasants’ breasts along the bone. Remove and reserve the skin, slice the meat on the bias into ½-inch strips, then transfer the meat and skin to a bowl and refrigerate until ready to use. Rub the carcasses with 2 tablespoons of the oil, season generously with salt and black pepper, arrange on a large rimmed baking sheet, and bake until browned all over, 45–55 minutes. Set aside and turn off the oven.
2. Meanwhile, make the crust: In a food processor, pulse together the flour, thyme, fine salt, and sugar. Add the butter and pulse until sandy. Pour ¼ cup of ice water evenly over the top and pulse until the dough just comes together. (If it looks dry, gradually add up to 2 tablespoons more water.) Transfer to a lightly floured work surface and knead once or twice to bring the dough together. Shape into a disk, wrap in plastic, and refrigerate for at least 1 hour, or up to 24. (Remove the dough from the fridge 15 minutes before rolling.)
3. Transfer the roasted pheasant carcasses and any accumulated juices along with the reserved pheasant skin to a large pot. Add the leek and onion scraps, mushroom stems, and 2 quarts of cold water and bring to a boil. Turn the heat to medium-low and simmer until the meat pulls easily from the bones, about 2 hours. Place a fine-mesh strainer over a bowl and strain the stock. When cool enough to handle, pick the meat from the carcasses, discarding the bones and skin. Coarsely chop the meat and ­transfer to a large bowl.
4. Clean and dry the pot, add the remaining oil, and turn the heat to medium. When the oil is hot and shimmering, add the reserved breast meat and cook, stirring occasionally, until lightly browned, 6–8 minutes. Using tongs, transfer to the bowl with the picked meat.
5. To the pot, add the leeks and onion and cook, stirring occasionally and scraping up any browned bits, until softened, 6–8 ­minutes. Add the butter, mushrooms, and garlic and cook, stirring occasionally, until the mushrooms are softened and fragrant (do not brown), about 6 minutes. Stir in the flour and reserved meat and cook until no streaks of flour remain, 1–2 minutes. Stir in 1 cup of the parsley, the thyme, mustard, bay leaves, and 4 cups of the reserved stock and bring to a boil. Turn the heat to medium-low and simmer gently, stirring and scraping the bottom of the pot frequently, until the sauce thickens, 7–10 minutes. Turn the heat to medium-low, cover, and continue cooking (do not boil), stirring occasionally, until the vegetables are very tender and the flavors have melded, about 1 hour. (If the pot looks dry or begins to scorch, add a bit more stock.)
6. Add the cream and continue ­cooking until the sauce thickens, 15–20 ­minutes. Season to taste with salt and black pepper, scrape into a 3-quart baking dish (no more than 11 inches in diameter), and set aside to cool to room temperature.
7. Position a rack in the center of the oven and preheat to 400°F. In a small bowl, lightly beat the egg and cream. On a lightly floured work surface, roll the dough into a 12-inch round measuring ¼-inch thick. Drape the dough over the filling, trimming as needed and crimping decoratively if desired, then seal the pie tightly. Brush with the egg wash, cut a few vents in the top, and sprinkle with flaky salt. Transfer to a rimmed baking sheet and bake until the filling is ­bubbling and the crust is deep golden brown, 45–50 minutes. Set aside to cool for at least 10 minutes, then serve, sprinkled with the remaining parsley.