Make the crust dry mixture: In a large bowl combine 2 1/2 cups all-purpose flour, 1 tablespoon sugar, and 1 teaspoon salt. Add 1 cup cold unsalted butter, cut into cubes, and toss to coat the cubes in the flour.

Incorporate the butter: Dump the mixture onto a clean work surface. Using a rolling pin, press and roll the butter into the flour to form thin, flaky pieces of butter distributed through the flour. Use a bench scraper to gather the mixture back into a pile as needed. Continue until the mixture is very flaky and the butter is well distributed. Return the mixture to the bowl and place it in the freezer for 15 minutes to chill the butter.

Bring the dough together: Remove the bowl from the freezer. Add 1/2 cup cold buttermilk and stir with a spoon, then use your hands to bring the dough together until it forms a cohesive ball. If the dough feels too dry, add 1 to 2 tablespoons cold water, 1 tablespoon at a time, only as needed. Divide the dough into two equal portions, flatten each into a disk, wrap each disk in plastic wrap, and chill in the refrigerator while you make the filling (chill at least while you prepare the filling; about 20–30 minutes).

Make the filling base: In a large skillet over medium-high heat melt 1/4 cup unsalted butter. Add 1/3 cup diced onion, 2 medium carrots sliced (about 1 cup), 1 stalk celery sliced (about 1/2 cup), and 2 cloves garlic minced. Cook, stirring occasionally, until the vegetables are tender, about 5–7 minutes.

Thicken the filling: Sprinkle 1/3 cup all-purpose flour over the cooked vegetables and stir to coat. Add 1 teaspoon salt, 1/2 teaspoon black pepper, 1 1/2 teaspoons minced fresh thyme, and 1 tablespoon minced fresh Italian parsley; stir to combine.

Add liquids and simmer: While whisking, slowly pour in 1 3/4 cups chicken broth and 1/2 cup heavy cream, whisking until the mixture is smooth and there are no lumps. Reduce the heat to medium-low and simmer, stirring occasionally, for about 10 minutes or until the sauce has thickened.

Finish the filling: Stir 3 cups shredded chicken or turkey and 1 cup frozen peas into the sauce. Remove the skillet from heat and set the filling aside while you roll out the dough.

Preheat the oven: Preheat the oven to 400°F.

Roll and fit the bottom crust: On a lightly floured surface, roll one chilled dough disk into a 12-inch circle about 1/4 inch thick. Carefully transfer the dough to a 9-inch pie pan, gently pressing it into the bottom and up the sides. Leave about 1/2 inch of overhang for sealing; trim any excess if there is a large overhang.

Fill the pie: Spoon the prepared filling into the pie shell and spread it evenly.

Top crust and seal: Roll the second dough disk into a 12-inch circle about 1/4 inch thick. Place it over the filling. Trim excess dough, leaving about 1/2 inch overhang. Fold the top and bottom edges together and crimp to seal using your fingers or a fork. Cut a few small slits in the center of the top crust to vent steam.

Apply egg wash: Brush the top crust and edges with the beaten egg (1 large egg, beaten).

Bake: Bake the pie at 400°F for about 45 minutes, or until the crust is golden brown. If the edges brown too quickly, protect them with a pie crust shield or foil.

Rest and serve: Remove the pie from the oven and cool for 10 minutes before cutting into slices and serving.