01 - Preheat the oven to 400°F (200°C).
02 - In a large skillet, melt butter over medium heat. Add onion and celery; sauté for 3-4 minutes until softened and translucent.
03 - Add flour to the skillet and cook, stirring constantly, for 1-2 minutes to form a smooth roux.
04 - Gradually whisk in chicken broth and milk. Cook, stirring continuously, until the sauce thickens, about 5 minutes.
05 - Stir in the potatoes, carrots, peas, chicken, salt, pepper, thyme, and parsley. Simmer for 8-10 minutes until the vegetables are just tender.
06 - Pour the filling into a 9-inch pie dish or deep baking dish.
07 - Roll out the puff pastry and lay it over the filling, trimming any excess. Press edges to seal and cut a few slits in the top to allow steam to escape.
08 - Brush the crust evenly with beaten egg for a golden finish.
09 - Bake for 35-40 minutes, or until the crust is golden brown and the filling is bubbling.
10 - Let cool for 10 minutes before serving to allow the filling to set.