Preheat the oven to 375°F.
Place potatoes in a large pot and cover with water. Bring to a boil for 10 minutes, then add carrots, peas, and corn and boil for 15 minutes more. Remove from the heat, drain the water and set aside.
Place bottom crust in a 9-inch pie plate and bake for 5 minutes.
While the crust is baking, cook butter in a saucepan over medium heat. Once butter is melted, stir in flour, garlic salt, and pepper. Slowly whisk in chicken broth and milk and heat until thickened, 5–6 minutes. Add turkey and veggies and mix together.
Remove bottom crust from the oven and pour the chicken broth mixture into the crust.
Cover with the top pie crust, making sure to seal the edges. Cut four (2-inch) slits on the top with a sharp knife to allow steam to escape.
Create an egg wash by whisking the egg with a little water. Brush egg wash on top the pie crust.
Bake for 40–45 minutes, or until the top is golden brown.
Let rest for at least 30 minutes before cutting and serving.