01 - Set oven temperature to 400°F and allow it to reach full temperature.
02 - Melt butter in a large skillet over medium heat. Add onion, carrot, and celery, sautéing for 5 minutes until vegetables are softened.
03 - Sprinkle flour over the softened vegetables and stir constantly for 2 minutes to form a paste base.
04 - Gradually whisk in chicken broth and milk, bringing to a simmer while stirring continuously until sauce thickens, approximately 4 to 5 minutes.
05 - Add cooked chicken, peas, thyme, parsley, salt, and pepper to the sauce. Stir thoroughly and cook for 2 minutes. Remove from heat.
06 - Pour filling into a 9-inch pie dish or divide equally among 4 individual ramekins.
07 - Roll out puff pastry sheet to fit over the dish. Drape pastry over filling, trim excess, and press edges firmly to seal.
08 - Cut small slits in pastry surface to allow steam escape. Brush entire surface with beaten egg.
09 - Bake for 25 to 30 minutes until pastry achieves golden brown color and crisp texture.
10 - Allow dish to stand 5 minutes before serving to allow filling to set slightly.