Preheat oven to 375 degrees
In a medium to large saucepan over medium heat, add the butter, onions and celery and cook until soft and translucent.
Stir in flour, salt and pepper.
Add the chicken broth. milk frozen carrots & peas, and cubed chicken. Simmer over medium-low heat until thick. About 5 to 10 minutes. Taste for flavor and add more pepper and/or salt if needed.
Remove from heat and set aside to cool off and continue to thicken. Let it rest for at least 30 minutes.
Line the bottom of a 9-in. pie pan with one of the pie crust. See Note 1 below.
Pour the chicken mixture over the bottom pie crust. Cover with top crust and seal edges, Make several small slits in the top to allow steam to escape.
Optional: Brush the top with egg wash.
Place the pie pan on a baking tray so that it's easy to remove from the oven once it's baked. Bake for about 40 to 45 minutes or until pastry is golden brown.
Cool for 10 minutes before serving.