01 - Set oven temperature to 400°F.
02 - Melt butter in a large skillet over medium heat. Add diced onions, carrots, and celery; cook for 5 to 6 minutes until softened.
03 - Stir in minced garlic and cook for 1 minute until aromatic.
04 - Sprinkle flour over vegetables and stir constantly for 1 to 2 minutes.
05 - Gradually whisk in chicken or turkey broth, whole milk, and heavy cream; bring mixture to a gentle simmer and cook until thickened, about 5 minutes.
06 - Add diced turkey, frozen peas, chopped parsley, dried thyme, salt, and black pepper. Stir well and simmer for 2 to 3 minutes, then remove from heat.
07 - Transfer filling to a 9-inch deep pie dish.
08 - Roll puff pastry sheet to fit over the dish. Drape over filling, trim excess dough, and press edges to seal. Cut small slits in pastry to vent steam.
09 - Brush puff pastry surface evenly with beaten egg.
10 - Place pie in oven and bake for 30 to 35 minutes until pastry is golden and filling bubbles.
11 - Allow pie to rest for 10 minutes to set before slicing.