Preheat the oven to 425 degrees. Line a large baking sheet with foil and set aside. For this recipe, use a 9-inch deep dish pie plate. Remove pie crusts from the refrigerator and allow them to come to room temperature.
Melt the butter in a large deep skillet or Dutch oven over medium-high heat, add the potatoes and cook them just a couple of minutes (you don’t want to cook them all the way or they will be too soft in the pie).
Then add the onion, carrots, celery, and peas. Stir them around until the onions start to turn translucent, about 3 minutes. Add the garlic and let cook for about 30 seconds.
Stir in the chicken or turkey and then sprinkle the flour over the top and stir it until it’s all combined with the turkey and vegetables. Cook for 1 minute.
Then pour in the chicken broth or stock. Stir to incorporate everything and let it cook and thicken. Once it starts to thicken salt, pepper, thyme, and rosemary. Mix well.
Add the half-and-half cream, then stir the mixture and let it bubble and thicken, about 3 minutes. If it seems too thick, drizzle in a little more broth. Turn off the heat. (Note: You want this filling on the thick side so it won’t seep through the bottom crust. Too runny of a filling and you risk it becoming a soggy bottom crust.)
Unroll one of the pie crusts and place it into a 9-inch, ungreased deep dish, glass pie plate. Gently press the crust down into the bottom and edges of the pie plate.
Pour the hot filling into the crust.
Top with the remaining crust, tucking the top crust up under the edges of the bottom crust. Flute the edges and cut vents into the top of the crust.
Mix together the egg with 2 tablespoons of water and brush it all over the surface of the crust. (You will have some egg wash left over.)
Place the pie pan on a prepared baking sheet (to catch any drippings). Bake immediately at 425 degrees for 15 minutes. At this point, if needed, loosely place a sheet of foil around the edges of the crust to prevent over-browning. Continue baking the pot pie for an additional 15 to 25 minutes, or until the crust is golden brown and the filling is bubbling. Let stand for 10 minutes before cutting.