01 - Preheat oven to 400°F and lightly grease a 9x13-inch baking dish.
02 - Melt butter in a large skillet over medium heat. Add onion, carrots, and celery; sauté 5 to 6 minutes until softened.
03 - Stir in flour and cook, stirring constantly, for one minute to form a roux.
04 - Gradually whisk in chicken broth and milk. Add salt, pepper, thyme, and garlic powder. Simmer, stirring frequently, until thickened, about 3 to 4 minutes.
05 - Fold shredded chicken and peas into the sauce mixture. Remove from heat.
06 - Transfer filling evenly into prepared baking dish.
07 - Unroll pie dough and cover the filling, trimming excess. Cut small slits in crust to allow steam to escape.
08 - Brush beaten egg evenly over the crust surface to enhance browning.
09 - Bake for 35 to 40 minutes until crust is golden and filling bubbles.
10 - Allow casserole to rest for 10 minutes to set before serving.