01 - Preheat oven to 400°F. Lightly grease a 9x13-inch baking dish.
02 - In a large skillet over medium heat, melt butter. Add onion, carrots, and celery; sauté 5–6 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 sauce thickens, about 3–4 minutes.
05 - Fold in shredded chicken and peas. Remove from heat.
06 - Transfer mixture evenly into prepared baking dish.
07 - Unroll pie dough over filling, trimming excess. Cut small slits in crust to vent.
08 - Brush crust with beaten egg for a golden finish.
09 - Bake 35–40 minutes until crust is golden brown and filling is bubbling.
10 - Allow casserole to rest 10 minutes before slicing and serving.