Preheat your oven to 375°F (190°C).
In a large skillet, melt the butter over medium heat. Sauté diced carrots, celery, and onions until tender.
Sprinkle flour over the vegetables and stir to coat. Cook for 1-2 minutes to remove the raw flour taste.
Gradually whisk in chicken broth and milk, stirring constantly to form a creamy sauce. Let simmer until thickened.
Season the sauce with salt, pepper, and your choice of fresh herbs like thyme or parsley.
Fold in shredded chicken and frozen peas. Stir until well combined.
Pour the filling into a pie dish lined with the bottom pie crust.
Cover with the top crust, crimp the edges, and cut slits to allow steam to escape.
Brush the top crust with a beaten egg or milk for a golden finish.
Bake for 40-45 minutes, or until the crust is golden brown and the filling is bubbly.
Let cool for 5-10 minutes before serving.