In a pot, combine chicken, carrots, peas, potatoes and celery. Add the chicken broth and boil for 15 minutes. Remove from heat, drain (reserving 1 and 3/4 cups chicken broth for filling) and set aside.
Place the bottom crust on the pie pan and dab a few times with a fork. Place in oven and cook at 425°F for 5-7 minutes.
In a large sauce pan, cook onions in butter (over medium-heat) until soft and translucent. Stir in flour, salt, pepper and celery seed. Next, slowly stir in chicken broth remains and milk. Simmer over medium-low heat until thick. Remove from heat and combine with the chicken, carrots, peas and celery mixture.
Remove pie crust from oven and turn temperature down to 375°F. Place the chicken filling in the bottom crust. Cover with top crust, seal edges and cut away excess dough. Make several small slits in the top to allow steam to escape.
Bake for 25 to 30 minutes, or until golden brown. Cool for 10 minutes before serving.