01 - Preheat the oven to 400°F.
02 - In a large skillet, melt butter over medium heat. Add onions, carrots, celery, and potatoes. Sauté for 8 to 10 minutes until softened.
03 - Stir in flour and cook for 1 to 2 minutes while stirring constantly.
04 - Gradually whisk in broth, then milk. Cook and stir until the mixture thickens, about 4 to 5 minutes.
05 - Add salt, pepper, thyme, and sage. Stir in turkey and peas. Simmer for 2 to 3 minutes, then remove from heat.
06 - Pour the filling into a 9-inch pie dish.
07 - Place puff pastry sheet over the filling, trim excess, seal edges, and cut small slits on top for steam escape.
08 - Brush the puff pastry with the beaten egg to achieve a glossy finish.
09 - Bake for 30 to 35 minutes until crust is golden and filling bubbles.
10 - Allow to stand for 10 minutes before serving.