01 - Preheat oven to 400°F.
02 - Melt butter in a large skillet over medium heat. Add diced onion, carrots, and celery, and sauté for 5 to 6 minutes until softened.
03 - Add minced garlic and cook for 1 minute until fragrant.
04 - Sprinkle flour over vegetables and cook, stirring constantly, for 1 to 2 minutes to form a roux.
05 - Gradually whisk in broth, whole milk, and heavy cream. Bring to a gentle simmer and cook until thickened, about 5 minutes.
06 - Stir in diced turkey, peas, parsley, thyme, salt, and pepper. Simmer for 2 to 3 minutes then remove from heat.
07 - Pour filling into a 9-inch deep pie dish. Roll out puff pastry to cover the dish, drape over filling, trim excess, and press edges to seal. Cut slits in the pastry for steam release.
08 - Brush the puff pastry surface with beaten egg to achieve a golden finish.
09 - Bake in preheated oven for 30 to 35 minutes until pastry is golden brown and filling bubbles.
10 - Allow to stand for 10 minutes before serving to set the filling.