Preheat your oven to 400°F (200°C) and lightly grease a 9-inch pie dish.
Melt the butter in a large skillet over medium heat. Add the onion, carrots, celery, and potatoes. Cook for 6 to 8 minutes, stirring occasionally, until the vegetables begin to soften. Add the minced garlic and cook for another 30 seconds.
Sprinkle the flour over the vegetables and stir well. Cook for about 1 minute to remove the raw flour taste. Slowly pour in the chicken broth while stirring constantly. Add the milk and continue stirring until the sauce begins to thicken.
Stir in the cooked chicken, frozen peas, thyme, parsley, salt, and black pepper. Cook for another 3 to 5 minutes, until the filling is thick and creamy. Remove the skillet from the heat.
Pour the chicken filling into the prepared pie dish. Place the pie crust over the top and gently press the edges around the dish. Trim excess crust and cut a few small slits in the center to allow steam to escape.
Beat the egg with 1 tablespoon water. Brush the mixture evenly over the top crust to help create a beautiful golden-brown finish.
Place the pot pie on a baking sheet. Bake for 30 to 35 minutes, or until the crust is golden brown and the filling is bubbling around the edges. If the crust browns too quickly, loosely cover the edges with aluminum foil.
Let the pot pie rest for 10 to 15 minutes before serving. This gives the creamy filling time to set slightly and makes it easier to scoop.