01 - Preheat oven to 400°F (200°C). Position the rack in the center of the oven.
02 - In a large skillet, melt the butter over medium heat. Add the diced onion, carrots, and celery. Sauté for 5 to 7 minutes until the vegetables are softened and the onions are translucent. Add the minced garlic and cook for 1 additional minute until fragrant.
03 - Sprinkle the flour over the sautéed vegetables and stir constantly for 2 minutes to cook out the raw flour taste. Slowly pour in the chicken broth while whisking continuously to prevent lumps. Add the whole milk and continue whisking until the sauce thickens and becomes smooth, about 4 to 5 minutes.
04 - Season the sauce with salt, black pepper, and dried thyme. Fold in the diced cooked chicken and frozen peas, stirring gently until evenly distributed throughout the filling. Remove the skillet from heat.
05 - Roll out one sheet of pastry to fit a 9-inch deep pie dish or medium casserole dish. Carefully lay the pastry into the dish, pressing it gently against the bottom and sides. Trim any excess dough hanging over the edges.
06 - Pour the chicken and vegetable filling into the pastry-lined dish, spreading it into an even layer. Cover with the second sheet of pastry. Trim the excess dough, then crimp the edges firmly together to seal. Cut 3 to 4 small slits in the top crust to allow steam to escape during baking.
07 - Brush the entire top surface of the pastry evenly with the beaten egg to ensure a rich golden-brown finish.
08 - Place the pie dish on the center rack of the oven. Bake for 35 to 40 minutes until the crust is deeply golden brown and the filling is bubbling through the steam vents.
09 - Remove from the oven and allow the pot pie to rest for 10 minutes before slicing. This resting period lets the filling set slightly, making it easier to serve.