01 - Preheat oven to 400°F. Lightly grease a 9x13-inch baking dish.
02 - Melt butter in a large skillet over medium heat. Add onion, carrots, and celery; sauté for 5 to 6 minutes until softened.
03 - Stir in flour and cook, stirring constantly, for 1 minute to form a roux.
04 - Gradually whisk in chicken broth and milk. Add salt, pepper, thyme, and garlic powder. Simmer, stirring, until thickened, about 3 to 4 minutes.
05 - Fold shredded chicken and peas into the sauce mixture. Remove from heat.
06 - Transfer filling to prepared baking dish, spreading evenly.
07 - Unroll pie dough over filling, trim excess, and cut small slits for steam vents.
08 - Brush pie dough with beaten egg to ensure a golden crust.
09 - Bake for 35 to 40 minutes until crust is golden and filling bubbles.
10 - Allow casserole to rest 10 minutes before serving.