Preheat oven to 425 degrees F (220 degrees C). Line a pie plate with 1 of the pie crusts.

Combine chicken, carrots, peas, and celery in a pot; pour in enough water to cover. Bring to a boil and cook until chicken is cooked through and vegetables are tender, about 15 minutes. Drain water and set chicken and vegetables mixture aside.

Melt butter in a saucepan over medium heat. Cook and stir onion in hot butter until soft and translucent, 7 to 10 minutes. Stir flour, salt, pepper, and celery seed into onion mixture until well-combined. Slowly stir chicken broth, celery soup, and milk into onion mixture; bring to a simmer, reduce heat to medium-low, and simmer, stirring occasionally, until mixture is thick, 5 to 10 minutes.

Spread chicken mixture in the bottom of the prepared pie crust. Pour hot onion mixture over chicken mixture. Lay remaining pie crust over the top, trim excess dough, and seal the edges. Cut several slits in the top crust to allow steam to escape.

Bake in the preheated oven until crust is golden brown and filling is bubbly, 30 to 35 minutes. Cool for 10 minutes before slicing.