01 - Preheat the oven to 400°F.
02 - Bring salted water to a boil in a medium saucepan. Add carrots, potatoes, and celery, and cook for 6 to 8 minutes until just tender. Drain and set aside.
03 - In a large skillet over medium heat, melt butter. Add onions and sauté for 3 to 4 minutes until translucent.
04 - Stir in flour and cook for 1 to 2 minutes, stirring constantly to form a roux.
05 - Gradually whisk in broth then milk until smooth. Bring to a simmer and cook for 3 to 4 minutes until thickened.
06 - Add thyme, sage, salt, and pepper. Stir in turkey, cooked vegetables, and peas. Simmer for 2 to 3 minutes.
07 - Transfer the filling to a 9-inch deep-dish pie plate or baking dish.
08 - Roll puff pastry to fit the dish. Place over filling, trim excess, gently press edges to seal, and cut slits on top for venting.
09 - Brush the pastry surface with beaten egg.
10 - Bake for 30 to 35 minutes until crust is golden and filling bubbles.
11 - Let stand for 10 minutes before serving.