Preheat your oven to 375°F.
Melt 1/2 cup butter in a large skillet over medium heat.
Add the carrots, potatoes, onions, and celery, cooking them until softened.
Add the flour to the skillet, stirring it in with the softened vegetables to create a thick paste.
Slowly whisk in the chicken broth and milk, letting the mixture simmer until it thickens.
Stir in the diced chicken and frozen peas.
Flavor everything with salt and pepper to your liking.
Pour the filling into a pie plate lined with the pie crust.
Make sure to spread it evenly across the bottom.
Cover the filling with another layer of pie crust.
Place the pie in the preheated oven, baking it until the crust turns golden brown, approximately 35-45 minutes.
Let the pie rest for 10 minutes before serving.