01 - Preheat oven to 400°F.
02 - Melt butter in a large skillet over medium heat. Add onions, carrots, celery, potatoes, and garlic; sauté for 5-7 minutes until softened.
03 - Stir in flour and cook for 1-2 minutes, stirring constantly to form a roux.
04 - Gradually whisk in broth and milk until smooth. Simmer for 5-7 minutes until sauce thickens.
05 - Add salt, pepper, thyme, and sage. Stir in turkey and peas; cook an additional 2 minutes.
06 - Transfer filling to a 9-inch deep-dish pie plate or equivalent baking dish.
07 - Unroll thawed puff pastry and cover filling, trimming excess. Press edges to seal and cut slits on top for ventilation.
08 - Brush the pastry surface with beaten egg to achieve a golden finish.
09 - Bake for 30-35 minutes until pastry is golden and filling bubbles.
10 - Allow to rest for 10 minutes to set filling before slicing and serving.