01 - Set oven temperature to 400°F.
02 - In a large skillet or Dutch oven, melt butter over medium heat. Add onion, carrots, celery, potatoes, and garlic; cook for 5-7 minutes until softened.
03 - Stir in flour and cook for 1-2 minutes, stirring constantly to form a roux.
04 - Gradually whisk in broth and milk, stirring until smooth. Bring to a simmer and cook 5-7 minutes until thickened.
05 - Add salt, black pepper, thyme, and sage. Stir in turkey and peas, cooking an additional 2 minutes.
06 - Transfer filling to a 9-inch deep-dish pie plate or similar baking vessel.
07 - Unroll puff pastry over filling, trim excess, press edges to seal, and cut small vents on top.
08 - Brush the pastry surface with beaten egg to enhance browning.
09 - Bake for 30-35 minutes until the puff pastry is golden and filling bubbles.
10 - Allow to rest for 10 minutes prior to serving for optimal texture.