Prepare the Filling

1. Cook the Vegetables: In a large skillet, melt butter over medium heat. Add carrots, potatoes, and celery, and sauté for 5-7 minutes, until slightly softened.
2. Make the Roux: Stir in flour and cook for 1 minute, stirring constantly.
3. Add Liquids: Gradually whisk in chicken broth and milk, bringing the mixture to a simmer. Cook until thickened, about 3-5 minutes.
4. Combine Ingredients: Stir in chicken, peas, salt, pepper, and thyme. Remove from heat and let the filling cool slightly.

Assemble the Pot Pie

1. Preheat Oven: Preheat your oven to 425°F (220°C).
2. Prepare the Crust: Roll out one pie crust and fit it into a 9-inch pie dish, allowing the edges to hang slightly over the sides.
3. Add the Filling: Pour the cooled filling into the prepared crust.
4. Top with Second Crust: Place the second crust over the filling, trim the edges, and pinch to seal. Cut a few small slits in the top crust for ventilation.
5. Egg Wash (Optional): Brush the top crust with beaten egg for a glossy finish.

Bake

1. Place the pie dish on a baking sheet to catch any drips.
2. Bake for 30-35 minutes, or until the crust is golden brown and the filling is bubbling.
3. Let the pie cool for 10 minutes before serving.