Add the cubed 1 cup peeled potatoes and peeled and sliced 3/4 cup carrots to a large pot, and cover them with water. Cook over high heat until the water boils for several minutes and potatoes and carrots are tender when poked with a fork, but don’t overcook. Drain water and set potatoes and carrots aside.
While the potatoes and carrots are cooking (or after they have cooked), in a large pot, combine the 1/2 cup (1 stick) unsalted butter and 1/3 cup diced onion over medium heat, while the butter melts, stir often, cooking the onions. When onions are soft, add the 1/2 cup all-purpose flour, 1 teaspoon salt, 1/2 teaspoon dried thyme and 1/4 teaspoons ground black pepper and mix well.
Slowly add the 1 1/2 cups chicken broth or turkey stock and 3/4 cup milk. Stir until the gravy has thickened.
Add the 2 cups cubed or shredded cooked turkey (or chicken), 3/4 cup frozen peas, 1/4 cup frozen corn, potatoes and carrots to the gravy and stir to combine.
Prepare your pie dish by lining it with an uncooked pastry crust. Pour the turkey and vegetable mixture into the pie shell. Cover with the top pie crusts, and let the top layer overlap the bottom crust, pinching seams to seal, then fluting the edge of the seam.
Crack an egg into a small bowl and beat it. Brush the beaten egg over the top, then with a sharp knife, cut some slits in the top of the pie, so it can vent steam.
Bake in a preheated oven at 425ºF for 35-40 minutes. Once cooked, let it cool for about 10 minutes before slicing and serving. Enjoy!