01 - Preheat the oven to 400°F (200°C).
02 - In a large skillet over medium heat, melt the butter. Add diced onion, carrots, and celery; sauté for 5 to 7 minutes until softened. Stir in minced garlic and cook for an additional minute.
03 - Sprinkle flour over the vegetables and cook, stirring constantly, for 2 minutes to form a roux.
04 - Gradually whisk in the broth followed by the milk. Bring mixture to a simmer and cook until thickened, about 4 to 5 minutes.
05 - Stir in the cooked turkey, peas, salt, pepper, and thyme. Remove the skillet from heat.
06 - Transfer the filling into a 9-inch pie dish or equivalent baking dish.
07 - Roll out the puff pastry sheet to fit the top of the dish. Place it over the filling, pressing edges to seal. Trim any excess pastry and cut small slits on top to vent steam.
08 - Brush the puff pastry surface with the beaten egg to enhance browning.
09 - Bake uncovered for 30 to 35 minutes or until the pastry is golden brown and crisp.
10 - Allow the pot pie to rest for 10 minutes before serving to set the filling.