01 - Preheat the oven to 400°F.
02 - Melt the butter in a large skillet over medium heat. Add onion, carrots, and celery; sauté until softened, approximately 5 to 6 minutes.
03 - Add flour to the skillet and cook while stirring constantly for 1 minute to form a roux.
04 - Gradually whisk in the chicken broth and milk. Cook, stirring frequently, until the mixture thickens and begins to bubble, about 3 to 4 minutes.
05 - Stir in cooked chicken, peas, salt, pepper, thyme, and garlic powder until evenly incorporated. Remove from heat.
06 - Transfer the filling to a 9-inch pie dish.
07 - Roll out pie dough and place over the filling. Trim excess dough and crimp edges to seal. Cut small slits on top for steam to escape.
08 - Brush the crust with the beaten egg to promote browning.
09 - Bake for 35 to 40 minutes until crust is golden and filling is bubbling.
10 - Allow to rest for 10 minutes prior to serving.