In a medium saucepan, add diced potatoes and carrots. Cover with water and bring to a boil.

Cook for about 10 minutes until tender but not mushy.

Drain and set aside.

In a large skillet or saucepan, melt the butter over medium heat. Add the diced onions and celery, and sauté until softened, about 5 minutes.

Stir in the flour and cook for 2 minutes to remove the raw flour taste.

Gradually whisk in the chicken or turkey broth until smooth. Let it simmer and thicken for about 3-5 minutes.

Stir in the milk or heavy cream, followed by the cooked turkey, peas, carrots, and potatoes. Add thyme, parsley, salt, and pepper to taste. Let the mixture simmer for a few more minutes until everything is well combined and heated through.

Remove from heat and let the filling cool slightly.

Preheat your oven to 400°F (200°C).

Roll out one sheet of pie crust and place it into a 9-inch pie dish, pressing it into the bottom and up the sides.

Pour the turkey and vegetable filling into the pie crust.

Roll out the second sheet of pie crust and place it over the top of the filling. Trim any excess dough and crimp the edges to seal. Cut a few slits in the top to allow steam to escape.

Brush the top crust with the beaten egg for a golden finish.

Bake in the preheated oven for 30-35 minutes, or until the crust is golden brown and the filling is bubbling.