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 continuously for 1 minute.
04 - Gradually whisk in chicken broth and milk. Add salt, black pepper, thyme, and garlic powder. Simmer, stirring, until sauce thickens, about 3 to 4 minutes.
05 - Fold shredded chicken and peas into the sauce mixture, then remove from heat.
06 - Transfer filling to prepared baking dish and spread evenly.
07 - Unroll pie dough and cover the filling, trimming excess. Cut small vent slits in topping.
08 - Brush beaten egg evenly over the crust surface.
09 - Bake 35 to 40 minutes until crust is golden and filling bubbles.
10 - Allow casserole to rest 10 minutes before slicing and serving.