Preheat the oven to 400 degrees F (200 degrees C).

Roll out 1 sheet of puff pastry into an 11-inch square. Press into a 9-inch pie dish, cut off extra pastry, and prick thoroughly with a fork. Press a sheet of aluminum foil right onto the pastry.

Bake in the preheated oven until pastry is puffed up and golden browned, about 25 minutes. Remove from oven, remove foil, and set aside.

Meanwhile, melt butter in a skillet over medium-low heat and cook onions and celery until soft and translucent, about 5 minutes. Add flour, salt, pepper, and celery seed. Cook and stir until mixture is paste-like.

Slowly add milk to the skillet while stirring constantly, about 5 minutes. Continue stirring and slowly add broth a little at a time. Cook until thickened, 7 to 10 minutes, constantly stirring. Add chicken and frozen vegetables. Cook until vegetables are thawed, 3 to 5 minutes.

Pour chicken mixture into pre-baked puff pastry shell. Roll out a second sheet of pastry into an 11-inch square and place on top of chicken mixture. Cut off extra pastry and crimp edges.