CRUST: 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. Transfer to a heatproof bowl and cool. 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 resembles coarse meal with some pea-size lumps. Drizzle with 3 tablespoons ice water. Pulse until just begins to hold together. (If mixture crumbles when squeezed, drizzle with 1 more tablespoon water and pulse a few more times.) Transfer to plastic wrap; knead just until dough holds together. Shape into a disk. Wrap tightly and refrigerate, about 1 hour.

FILLING: Brown butter for filling in pot. Add onions, celery, mushroom caps, and season with salt and pepper. Cook, stirring occasionally, until 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 squash and reserved broth and bring to a boil, then remove from heat. Tear chicken into bite-size pieces. Stir into pot with peas and parsley. Transfer to a 3-quart baking dish.

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.