Brown butter turns the crust for this comfort-food classic into a flaky, nutty marvel. Butternut squash and cremini mushrooms give the usual chicken-and-vegetable filling extra oomph.
Melt butter in a medium saucepan over medium-high. When it boils, reduce heat to medium; simmer until foamy. Continue cooking, stirring occasionally and scraping bottom of pan, until foam subsides, butter turns golden brown with a nutty aroma, and milk solids separate into brown specks that sink to bottom, 2 to 7 minutes. Remove from heat; immediately transfer to a heatproof bowl. Let cool completely. Pour into an ice-cube tray, filling cups halfway. Freeze until cold and firm, about 1 hour. Pulse flour and salt in a food processor to combine. Add chilled brown butter; pulse until mixture resembles coarse meal with some pea-size lumps. Drizzle with 3 tablespoons ice water. Pulse until mixture just begins to hold together. (If mixture crumbles when squeezed in your hand, drizzle with 1 more tablespoon ice water and pulse a few more times.) Transfer to a piece of plastic wrap; knead just until dough holds together. Loosely cover and shape into a disk. Wrap tightly and refrigerate until firm, about 1 hour.
Meanwhile, combine chicken, reserved leek tops, halved celery, mushroom stems, squash peelings, and 1 tablespoon salt in a large pot. Add broth and enough water tojust cover chicken. Bring to a boil, then reduce heat to medium and simmer until a thermometer inserted into thickest parts of chicken (without touching bones) registers 160 degrees, about 20 minutes. Transfer chicken to a plate. Strain broth; reserve 2 1/2 cups (refrigerate or freeze remainder for another use).
Brown butter for filling in pot. Add sliced leeks and chopped celery, mushroom caps, and squash; season with salt and pepper. Cook, stirring occasionally, until squash begins to soften, about 5 minutes. Stir in flour and thyme; cook until darkened slightly and nutty, about 2 minutes. Add wine; boil until mostly evaporated, about 30 seconds. Stir in reserved broth and bring to a boil, then remove from heat. Pull chicken from skin and bones and tear into bite-size pieces. Stir into pot with peas and parsley. Transfer to a 3-quart baking dish. (Filling can be made up to 2 days ahead and refrigerated; stir in peas and parsley just before baking.)
Preheat oven to 400 degrees. On a floured work surface, roll out dough to a diameter slightly larger than baking dish. Center dough over filling; trim edges to a 1/2-inch overhang. Beat egg white with 1 teaspoon water; brush over dough. Cut vents in center. Bake until crust is golden brown and filling bubbles, 35 to 45 minutes. Let cool 20 minutes before serving.
The dough can be made and refrigerated two days ahead -- and can also be frozen for up to six months (thaw completely before using).