Preheat your oven to 425°F (220°C).
In a large skillet or saucepan, melt the butter over medium heat. Add the chopped onion, carrots, and celery. Sauté for 5-7 minutes until the vegetables begin to soften.
Sprinkle the flour over the vegetables. Stir constantly for 1-2 minutes to cook off the raw flour taste.
Gradually whisk in the chicken broth and milk. Stir in the salt, pepper, and thyme. Simmer over medium-low heat for 5-8 minutes, stirring frequently, until the sauce thickens.
Remove the skillet from the heat. Stir in the cooked chicken and peas.
Place one pie crust into the bottom of a 9-inch pie plate. Pour the chicken filling into the crust.
Top with the second pie crust. Trim any excess dough, seal the edges by crimping with a fork or your fingers, and cut 3-4 slits in the top to allow steam to escape.
Brush the top crust with the beaten egg.
Bake for 30-40 minutes, or until the crust is golden brown and the filling is bubbly. (Tip: If the edges brown too quickly, cover them with foil strips/shield halfway through).
Let the pie cool for at least 15 minutes before slicing to allow the filling to set.