Prepare the Filling: In a saucepan, combine cubed chicken, carrots, peas, and celery. Add enough water to cover the ingredients and bring to a boil. Boil for 15 minutes, then drain and set aside.

Make the Sauce: In a separate saucepan, melt butter over medium heat. Add chopped onion and cook until soft, about 5-7 minutes. Stir in flour, salt, pepper, and celery seed, and cook for 1-2 minutes. Gradually add chicken broth and milk, whisking constantly until thickened, 3-5 minutes.

Assemble the Pie: Preheat the oven to 425°F (220°C). Roll out one pie crust and place it in a 9-inch pie dish. Add the chicken and vegetable mixture, then pour the creamy sauce over it. Roll out the second crust and place it over the filling. Seal edges by crimping and make small slits on top to allow steam to escape.

Bake the Pie: Bake for 40-50 minutes, or until the crust is golden and filling is bubbling. If edges brown too quickly, cover with aluminum foil and continue baking.

Cool and Serve: Let the pie cool for 10 minutes before serving. This allows the filling to set, making it easier to slice. Serve warm.