01 - Preheat the oven to 400°F. Position the rack in the center of the oven.
02 - In a large skillet, melt the butter over medium heat. Add the chopped onion and sliced celery, sautéing for 3 to 4 minutes until softened and translucent.
03 - Sprinkle the flour over the softened vegetables and cook, stirring constantly, for 1 to 2 minutes to form a smooth roux and cook out the raw flour taste.
04 - Gradually whisk in the chicken broth and whole milk, stirring continuously to prevent lumps. Cook for about 5 minutes until the sauce thickens and coats the back of a spoon.
05 - Stir in the diced potatoes, frozen carrots, frozen peas, diced chicken, salt, black pepper, dried thyme, and dried parsley. Simmer for 8 to 10 minutes until the vegetables are just tender.
06 - Pour the chicken and vegetable filling into a 9-inch pie dish or deep baking dish, spreading it evenly.
07 - Roll out the puff pastry and drape it over the filling. Trim any excess dough around the edges, press firmly to seal, and cut a few small slits in the top to allow steam to escape during baking.
08 - Brush the entire surface of the pastry crust evenly with the beaten egg to promote a rich golden-brown finish.
09 - Bake for 35 to 40 minutes until the crust is deep golden brown and the filling is bubbling through the steam vents.
10 - Remove from the oven and let the pot pie rest for 10 minutes before serving to allow the filling to set slightly.