Preheat oven to 425 degrees F
In a saucepan, combine chicken, carrots, peas, and celery. Add water to cover and boil for 15 minutes. Remove from heat, drain and set aside.
In the saucepan over medium heat, cook onions in butter until soft and translucent. Stir in flour, salt and pepper. Slowly stir in chicken broth and milk.
Simmer over medium-low heat until thick. Remove from heat and set aside.
Add chicken mixture to liquid mixture and stir.
Place 1 pie crust in bottom of pie pan and add chicken/veggie mixture
Cover with top crust, seal edges.
Make several small slits in the top to allow steam to escape.
Bake in the preheated oven for 30 to 35 minutes, or until pastry is golden brown and filling is bubbly.
Cool for 10 minutes before serving with a green salad or just on it's own!