2Heat butter in a saucepan over medium heat; cook and stir onion until soft and translucent, 5 to 10 minutes. Sprinkle flour, salt, black pepper, and celery seed over onion; stir until coated. Slowly pour broth and milk over onion mixture, stirring constantly. Bring mixture to a simmer over medium-low heat and cook until thickened, about 10 minutes. Remove saucepan from heat.
3Press 1 pie crust into a 9-inch pie dish. Fill crust with chicken, corn, green beans, and carrots. Pour hot liquid mixture over filling. Cover pie with remaining pie crust, seal edges, and cut away excess dough. Cut several slits into the top crust for ventilation during baking.
4Bake in the preheated oven until crust is golden brown and filling is bubbling, 30 to 35 minutes. Cool for 10 minutes before serving.