Preheat oven to 400°F (200°C).
Chop or shred the cooked turkey into bite-sized pieces.
Melt butter in a large skillet over medium heat.
Add onion, carrots, and celery and cook for 6–8 minutes until beginning to soften.
Add garlic and cook for 30 seconds.
Sprinkle flour over the vegetables and stir for 1–2 minutes.
Gradually stir in the broth until smooth.
Add milk and cook for 3–5 minutes until the sauce thickens.
Stir in thyme and black pepper. Taste and add salt as needed.
Fold in the cooked turkey and frozen peas.
Remove from heat and transfer filling to a 9-inch deep pie dish or approximately 2-quart baking dish.
Let the filling cool for 10–15 minutes.
Place cold puff pastry over the baking dish without stretching it.
Trim excess pastry if necessary and lightly press around the rim.
Cut 3–4 small steam vents in the center.
Whisk egg with 1 tablespoon water and lightly brush over the pastry.
Place the dish on a baking sheet.
Bake for 25–35 minutes until the pastry is deeply golden and the filling bubbles around the edges.
Rest for 10 minutes before serving.