01 - Preheat the oven to 400°F (200°C).
02 - In a large skillet, melt the butter over medium heat. Add the onion and celery; sauté for 3-4 minutes until softened and translucent.
03 - Sprinkle in the flour and cook, stirring constantly, for 1-2 minutes to form a smooth roux and cook out the raw flour taste.
04 - Gradually whisk in the chicken broth and whole milk. Continue cooking, stirring frequently, until the sauce thickens and coats the back of a spoon, about 5 minutes.
05 - Fold in the diced potatoes, frozen carrots, frozen peas, diced chicken, salt, black pepper, dried thyme, and dried parsley. Simmer gently for 8-10 minutes until the vegetables are just tender.
06 - Pour the hot 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, press the edges 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 deep golden-brown finish.
09 - Bake on the center rack for 35-40 minutes, or until the crust is puffed and golden brown and the filling is bubbling through the steam vents.
10 - Remove from the oven and let cool for 10 minutes before serving to allow the filling to set slightly.