In a large skillet, melt the butter over medium heat. Add the chopped onion and celery, and sauté until they are soft and translucent, about 5-7 minutes.
Stir in the flour and cook for another minute to create a roux.
Slowly whisk in the chicken broth and milk, ensuring there are no lumps.
Add the garlic powder, thyme, and season with salt and pepper. Bring the mixture to a gentle simmer and let it thicken for about 5 minutes.
Once thickened, stir in the cooked chicken, mixed vegetables, and boiled potatoes. Remove from heat and set aside.
Roll out one pie crust and fit it into a 9-inch pie dish. Pour the chicken filling into the crust.
Roll out the second pie crust and place it on top of the pie. Crimp the edges to seal and make slits for 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 bubbly.
Remove from the oven and let it cool for 10 minutes before slicing.
Notes
Let the pie cool for 10 minutes before slicing to allow the filling to set.