Preheat oven to 425 degrees F. In a large saucepan, combine cubed chicken, sliced carrots, frozen peas, and celery. Cover completely with cold water, bring to a boil over medium-high heat, and cook for 15 minutes until chicken is cooked through and carrots are just tender. Drain thoroughly and set aside.

In a separate saucepan over medium heat, melt the butter and add the chopped onion. Cook for 3 to 4 minutes until soft and translucent. In a small bowl, whisk together flour, salt, pepper, and celery seed. Stir into the butter and onion mixture and cook for 1 minute.

Slowly pour in the chicken broth while whisking constantly to prevent lumps, then add the milk. Stir over medium-low heat for 5 to 7 minutes until the sauce thickens enough to coat the back of a spoon. Remove from heat.

Press one pie crust firmly into a 9-inch pie pan. Spread the drained chicken and vegetable mixture evenly over the bottom crust. Pour the hot cream sauce over the filling so it fills all the gaps.

Place the second crust over the top, press and crimp the edges to seal, and trim any excess overhang. Cut 4 to 5 small slits across the top crust to allow steam to escape during baking.

Bake at 425 degrees F for 30 to 33 minutes until the top crust is deep golden brown and the filling is visibly bubbling through the steam slits. Cover the edges loosely with foil strips if they brown too quickly before the center is done.

Remove from the oven and let the pie rest for at least 10 minutes before slicing to allow the filling to set.