01 - Preheat oven to 400°F.
02 - In a large skillet over medium heat, melt the butter. Add onion, carrot, and celery. Sauté for 5 minutes until softened.
03 - Sprinkle in the flour and stir constantly for 2 minutes to form a roux.
04 - Gradually whisk in the chicken broth and milk. Bring to a simmer, stirring until thickened, approximately 4 to 5 minutes.
05 - Add cooked chicken, peas, thyme, parsley, salt, and pepper. Stir and cook for 2 minutes. Remove from heat.
06 - Spoon the filling into a 9-inch pie dish or divide among 4 individual ramekins.
07 - Roll out the puff pastry sheet to fit over the dish. Drape pastry over the filling, trimming any excess. Press edges to seal.
08 - Cut a few small slits in the pastry to allow steam to escape. Brush with beaten egg.
09 - Bake for 25 to 30 minutes, or until the pastry is golden brown and crisp.
10 - Let stand 5 minutes before serving.