Preheat the oven to 425°F (220°C).
In a large skillet over medium heat, melt the butter. Add the diced onion, celery, and carrots, and sauté until softened, about 5 minutes.
Stir in the flour, thyme, garlic powder, salt, and pepper. Cook for 1-2 minutes until the mixture is bubbly.
Gradually whisk in the chicken broth and milk. Continue to cook, stirring frequently, until the mixture thickens, about 5-7 minutes.
Remove from heat and stir in the cooked chicken and frozen mixed vegetables. Mix until well combined.
Roll out one pie crust and fit it into a 9-inch pie dish. Pour the chicken mixture into the crust.
Roll out the second pie crust and place it over the filling. Seal the edges by crimping with your fingers or a fork. Cut a few slits in the top crust to allow steam to escape.
Brush the top crust with the beaten egg for a golden finish.
Bake in the preheated oven for 30-35 minutes, or until the crust is golden brown. Let it cool for 10 minutes before serving.