01 - Set the oven temperature to 400°F and allow it to fully preheat.
02 - Melt butter in a large skillet over medium heat. Add diced onion, carrots, and celery; cook for 5 to 7 minutes until softened. Stir in minced garlic and cook for an additional minute.
03 - Sprinkle flour over the vegetables and stir continuously for 2 minutes to cook the flour.
04 - Slowly whisk in the chicken or turkey broth followed by milk. Bring to a gentle simmer and cook for 4 to 5 minutes until the sauce thickens.
05 - Incorporate diced turkey, peas, salt, black pepper, and dried thyme into the sauce. Remove from heat once evenly mixed.
06 - Transfer the filling into a 9-inch pie dish. Roll out puff pastry and cover the filling. Seal the edges by pressing firmly and trim any excess dough. Cut small slits in the top to allow steam escape.
07 - Brush the puff pastry thoroughly with the beaten egg to ensure a golden crust.
08 - Place pie in the oven and bake for 30 to 35 minutes until the pastry is crisp and golden brown.
09 - Allow the pie to rest for 10 minutes after baking to set before slicing.