01 - Preheat oven to 400°F.
02 - Melt butter in a large skillet or Dutch oven over medium heat. Add onions, carrots, celery, potatoes, and garlic. Sauté for 5–7 minutes until softened.
03 - Sprinkle flour over vegetables and cook, stirring constantly, for 2 minutes.
04 - Gradually whisk in broth and milk. Bring to a simmer, stirring constantly, until thickened, about 5 minutes.
05 - Stir in turkey, peas, salt, pepper, thyme, and sage. Simmer for 2–3 minutes. Taste and adjust seasoning as needed.
06 - Transfer filling to a 9-inch deep-dish pie pan or ovenproof dish.
07 - Roll out puff pastry slightly to fit over the dish. Place pastry over filling, trimming edges as needed. Press pastry to edge of dish to seal.
08 - Cut a few small slits in the pastry for steam to escape. Brush with beaten egg.
09 - Bake for 25–30 minutes until pastry is golden and filling is bubbling.
10 - Let stand for 10 minutes before serving to allow filling to set.