01 - Preheat oven to 400°F and lightly grease a 9x13-inch baking dish.
02 - Melt butter in a large skillet over medium heat, then 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 whole milk, then add salt, pepper, thyme, and garlic powder; simmer while stirring until sauce thickens, about 3 to 4 minutes.
05 - Fold shredded chicken and frozen peas into the thickened sauce and remove from heat.
06 - Transfer filling into prepared baking dish, spreading evenly.
07 - Unroll pie dough over filling, trim excess, and cut slits to vent.
08 - Brush beaten egg evenly over the crust surface.
09 - Bake for 35 to 40 minutes until crust is golden and filling is bubbly.
10 - Allow casserole to rest for 10 minutes before serving.