01 - Preheat oven to 400°F.
02 - In a large skillet or Dutch oven, melt butter over medium heat. Add onions, carrots, celery, potatoes, and garlic. Sauté for 5–7 minutes, until vegetables are softened.
03 - Sprinkle flour over vegetables and cook, stirring, 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.
06 - Transfer filling to a 9-inch deep-dish pie pan or similar 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 10 minutes before serving.