Preheat the oven to 350 degrees F (175 degrees C).
Line a 9x13-inch casserole dish with 1 pie crust and cover with aluminum foil.
Bake crust in the preheated oven until golden, about 20 minutes.
Melt butter in a large skillet over medium heat; add celery, leeks, peas, carrots, potatoes, and onion. Cook and stir until onions are translucent and potatoes are tender, about 15 minutes. Add chicken and mix thoroughly.
Combine 1 cup of milk and flour in a cup. Mix well and add it to the vegetable-chicken mixture. Cook and stir, adding more flour if necessary to thicken gravy, about 5 minutes. Add remaining 1 cup milk; cook and stir until filling is thickened, about 5 minutes more. Season with black pepper, white pepper, and salt.
Pour filling into the baked crust. Roll out the remaining crust to the shape of the casserole dish and gently place it on top of the filling.
Preheat the oven to 450 degrees F (230 degrees C).
Bake in the preheated oven, checking every 10 minutes, until crust is light golden brown, about 30 minutes.