In a saucepan, combine chicken, carrots, peas, and celery. Add water to cover and boil for 15 minutes. Stir occasionally. Remove from heat, drain and set aside.
In the saucepan over medium heat, cook minced onions in butter until soft. Stir in flour, salt, pepper, and celery seed. Mix chicken broth and milk together. Slowly stir in chicken broth and milk mix. Simmer over medium-low heat until thick. Remove from heat and set aside.
Put pie crust in glass pie dish (must not be frozen) Place the chicken mixture in bottom pie crust. Pour hot liquid mixture over. Cover with top crust, seal edges, and cut away excess dough. Make several small slits in the top to allow steam to escape.
Brush pie crust with milk.
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.