In a large skillet, melt butter over medium heat. Add onions, carrots, celery, and mushrooms. Sauté for 5-6 minutes until softened. Add minced garlic and cook 1 more minute until fragrant.

Stir in flour, salt, pepper, and herbs. Cook 1-2 minutes to remove raw flour taste.

Slowly whisk in chicken broth and milk until smooth. Cook, stirring frequently, until the mixture thickens (about 3-5 minutes).

Stir in the cooked chicken and peas. Remove from heat and let the filling cool slightly.

Meanwhile, preheat your oven to 425°F (220°C).

On a lightly floured surface, roll out one pie crust to a12-inch circle and line a 9-inch pie dish. Pour in the filling.

Cover with the second crust, seal edges, and cut slits on top for steam to escape.

Lightly beat 1 egg with 1 tablespoon of water and brush it over the top crust. This gives the pie a beautiful golden-brown, glossy finish.

Bake in the preheated oven for 20 minutes. Reduce the oven temperature to 350°F and continue baking until the crust is golden brown and the filling is bubbly, about 35-40 minutes. If the edges brown too quickly, cover them with foil.

Let the pie cool 5-10 minutes before serving to allow the filling to set.