Preheat oven to 375 F

In a saucepan, combine chicken, carrots, green beans, and celery. Add water to cover and boil for 15 minutes. Remove from heat, drain and set aside.

In the saucepan over medium heat, cook onions in butter until soft and translucent. Stir in flour, salt, pepper, and celery seed. Slowly stir in chicken broth and mild. Simmer over medium-low heat until thick. Remove from heat and set aside.

Brush bottom of pie crust with egg white. Place the chicken mixture in bottom pie crust. Pour hot liquid mixture over. Cover with top crust, seal edges, and cut away excess dough. Brush top with remaining egg white and cut several small slits in the top to allow steam to escape.

Bake in preheated oven for 40-45 minutes, or until pastry is golden brown and filling is bubbly. Cool for 10 minutes before serving.