01 - Set the oven temperature to 400°F.
02 - Melt butter over medium heat in a large skillet. Add onion, carrots, and celery, cooking for 5 to 6 minutes until softened.
03 - Stir in minced garlic and cook for 1 minute until fragrant.
04 - Sprinkle flour over vegetables and cook, stirring constantly, for 1 to 2 minutes to eliminate raw taste.
05 - Gradually whisk in broth, whole milk, and heavy cream. Simmer gently until mixture thickens, about 5 minutes.
06 - Combine diced turkey, peas, parsley, thyme, salt, and pepper. Stir well and simmer for 2 to 3 minutes, then remove from heat.
07 - Transfer filling into a 9-inch deep pie dish or equivalent baking vessel.
08 - Roll out puff pastry to cover the dish. Place pastry over filling, trim excess, and press edges to seal. Cut small slits to allow steam to escape.
09 - Brush the surface of puff pastry with beaten egg to ensure a glossy, golden finish.
10 - Bake for 30 to 35 minutes until pastry is golden brown and filling bubbles.
11 - Allow to stand for 10 minutes to set before slicing and serving.