Preheat oven to 425 F. Line bottom of 9 inch pie dish with a pie crust, set aside.
In a large saucepan, combine chicken, carrots, peas, and celery. Add chicken broth and enough water to cover and boil for 15 minutes. Remove from heat, drain, reserving cooking liquid, and set aside.
In the saucepan over medium heat, melt butter and cook onions until soft and translucent. Stir in flour, salt, pepper, and celery seed and cook 2 to 3 minutes, making a white roux. Slowly whisk in 2 to 2-1/2 cups of reserved broth and milk. Whisk until smooth and flour mixture completely incorporated. Simmer over medium-low heat until thick. Remove from heat and set aside.
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 vents 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.