01 - Preheat the oven to 400°F.
02 - In a large skillet, melt butter over medium heat. Add diced onion, carrots, celery, and potatoes; cook, stirring occasionally, until softened, about 8 to 10 minutes.
03 - Sprinkle flour over the vegetables, stirring constantly, and cook for 1 to 2 minutes to eliminate raw flour taste.
04 - Slowly whisk in chicken or turkey broth, then whole milk. Continue stirring until the sauce thickens, approximately 4 to 5 minutes.
05 - Stir in salt, black pepper, dried thyme, and optional dried sage. Add cooked turkey and frozen peas, then simmer gently for 2 to 3 minutes before removing from heat.
06 - Transfer filling to a 9-inch pie dish. Lay the thawed puff pastry sheet on top, trimming any excess. Press edges to seal and cut several small slits on the surface to vent steam.
07 - Brush the puff pastry evenly with the beaten egg to ensure a golden crust.
08 - Bake in preheated oven for 30 to 35 minutes until the crust is golden brown and filling bubbles around edges.
09 - Allow to rest for 10 minutes to set before serving.