01 - Preheat oven to 400°F. Position rack in the center of the oven.
02 - In a large skillet, melt butter over medium heat. Add onions, carrots, and celery. Sauté for 5–7 minutes until softened. Add garlic and cook for 1 minute until fragrant.
03 - Stir in flour and cook for 2 minutes, stirring constantly to form a roux. Slowly whisk in chicken broth, then milk, and continue whisking until the mixture thickens, about 4–5 minutes.
04 - Add salt, pepper, thyme, cooked chicken, and peas. Stir to combine evenly. Remove from heat and set aside.
05 - Roll out one pastry sheet and line the bottom of a 9-inch deep pie dish or medium casserole dish. Pour in the chicken and vegetable filling, spreading it evenly.
06 - Cover with the second pastry sheet. Trim any excess dough and crimp the edges tightly to seal. Cut a few small slits in the top to allow steam to escape during baking.
07 - Brush the top pastry evenly with the beaten egg to ensure a rich golden finish.
08 - Bake for 35–40 minutes, or until the crust is deeply golden brown and the filling is bubbling through the steam vents.
09 - Let the pot pie rest for 10 minutes before slicing and serving. This allows the filling to set for cleaner portions.