01 - Preheat oven to 400°F. Position the rack in the center of the oven for even baking.
02 - Melt butter in a large skillet over medium heat. Add the chopped onion and sliced celery, sautéing for 3 to 4 minutes until softened and translucent.
03 - Sprinkle flour over the softened vegetables and stir constantly for 1 to 2 minutes, cooking until the mixture forms a smooth, lightly golden roux.
04 - Gradually whisk in the chicken broth and whole milk, stirring continuously to prevent lumps. Continue cooking for about 5 minutes until the sauce thickens enough to coat the back of a spoon.
05 - Fold in the diced potatoes, frozen carrots, frozen peas, diced chicken, salt, black pepper, dried thyme, and dried parsley. Simmer the mixture for 8 to 10 minutes until the vegetables are just tender.
06 - Pour the prepared filling into a 9-inch pie dish or deep baking dish, spreading it into an even layer.
07 - Roll out the puff pastry sheet and drape it over the filling. Trim any overhanging excess, press the edges firmly to seal against the dish, and cut a few small slits in the top to allow steam to escape during baking.
08 - Brush the pastry surface evenly with beaten egg to promote a rich golden-brown finish during baking.
09 - Bake for 35 to 40 minutes until the crust is deeply 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.