01 - Preheat oven to 400°F.
02 - In a large skillet over medium heat, melt butter. Add diced onion, carrots, and celery; sauté for 5 to 7 minutes until softened. Stir in minced garlic and cook for an additional minute.
03 - Sprinkle in flour and stir constantly for 2 minutes to create a roux.
04 - Gradually whisk in chicken or turkey broth followed by whole milk. Bring mixture to a simmer and cook until sauce thickens, about 4 to 5 minutes.
05 - Stir in diced turkey, frozen peas, salt, black pepper, and dried thyme. Remove from heat to prevent overcooking.
06 - Transfer filling into a 9-inch pie dish or equivalent baking vessel.
07 - Roll puff pastry sheet to fit over the dish. Place it atop the filling, pressing edges firmly to seal. Trim any excess and cut small slits on the surface to allow steam escape.
08 - Brush beaten egg evenly over the puff pastry surface to promote browning.
09 - Bake at 400°F for 30 to 35 minutes until the puff pastry is golden and crisp.
10 - Allow to rest for 10 minutes before serving to set the filling.