Make the pie crust through step 5 according to my directions and video tutorial in my pie crust recipe. Make pie dough in advance because it needs to chill in the refrigerator for at least 2 hours before rolling out (step 3). The recipe makes 2 crusts, and you only need 1 for this pie, so save the other for another pie.

Melt the butter in a large skillet or pot over medium heat. Add the onion, carrots, celery, mushrooms, and garlic. Cook, stirring occasionally, for 5 minutes or until vegetables have softened and released some liquid. Stir in flour, salt, pepper, thyme, and rosemary until flour has absorbed all the liquid. Stir in the broth and milk. Simmer for 7–8 minutes or until thickened into a gravy consistency. Stir in the frozen peas and turkey, and then remove from heat. Taste and add more salt & pepper, thyme, or rosemary to taste, if desired. Cool for 10 minutes at room temperature. (Or up to 1 day. Cover and store in the refrigerator if making 1 day ahead.) After it has cooled for at least 10 minutes, transfer the filling to a 9-inch pie dish.

Preheat oven to 375°F (191°C).

Remove 1 disc of pie dough from the refrigerator. On a lightly floured work surface using a lightly floured rolling pin, roll the dough out into an 11- or 12-inch circle. Make sure to turn the dough about a quarter turn after every few rolls. Carefully lay dough over top of the filling. Trim dough edges so there is only about an inch of overhang. Fold the overhanging dough under, melding it into itself to form a thick edge crust. Crimp the edges with a fork or flute the edges with your fingers.

Cut slits in the top to make steam vents. Lightly brush the top of the pie crust with the egg wash.

Bake for 35–40 minutes or until the top of the crust is golden brown. After 20 minutes of baking, be sure to cover the edges of the crust with aluminum foil or use a pie crust shield to prevent the edges from getting too brown. (See this post on the best pie baking tools for instructions on how to make a pie crust shield out of foil.)

Remove from the oven and allow to cool for 5 minutes before serving. Leftovers keep well in the refrigerator in an airtight container for up to 5 days. Reheat as desired.