Preheat oven to 425 degrees F.
In a large saucepan over medium heat, cook onions and celery in butter until they start to soften. Add cubed, uncooked chicken and sauté for about 8-10 minutes.
Mix together the milk and flour. Add to the chicken. Stir until it begins to thicken. Slowly add the broth and simmer over medium-low heat until thick. Stir in seasonings and carrots and peas. Adjust salt and pepper to taste. Remove from heat and set aside.
Line a 9-inch pie with an unbaked pie crust. Pierce the crust several times with the tines of a fork. Pour warm chicken filling into the crust. Cover with top crust, seal the edges, and cut away any excess dough. Make several small slits in the top to allow steam to escape.
Bake in the preheated oven for 30 to 35 minutes or until pie crust is golden brown and filling is bubbly. Cool for 10 minutes before serving.