Add carrots and celery to a medium saucepan. Fill with water and bring to a boil over high heat. Reduce the heat to medium and simmer 8 minutes. Add in peas and continue simmering for another 2 minutes. Drain.
Meanwhile, in another saucepan, melt butter over medium heat. Saute the onion in the melted butter for about 5 minutes until softened. Add in the flour and stir until well combined. Whisk in chicken broth and cook over medium heat until thickened, about 5 minutes.
Add the vegetable mixture to the sauce along with cooked, diced chicken. Season with salt and pepper.
Preheat oven to 425 degrees Fahrenheit. Line a deep dish pie plate with the prepared bottom crust, letting it hang over the edge by 1/2-inch.
Pour the pot pie filling onto the bottom crust. Cover with top crust and seal the edges. Using a sharp knife, slice several small slits in the top crust to allow the steam to escape while baking.
Bake pie in a 425 degree oven for 30-35 minutes, until the top crust is golden brown. Allow to cool for 10 minutes before serving.
MAKE AHEAD INSTRUCTIONS: To make chicken pot pie ahead, cook your filling and prepare your pie shells. Store separately in the fridge until ready to assemble and bake. FREEZER INSTRUCTIONS: Assemble your pot pie as directed in the recipe and freeze immediately. When ready to bake, preheat your oven to425 degrees. Bake, uncovered, for 15 minutes. Cover the pot pie with aluminum foil in order to prevent the crust from burning and continue baking for 30 to 35 minutes more, until center is hot.