01 - Set oven temperature to 400°F.
02 - Melt butter in a large skillet over medium heat. Add onion, carrots, and celery and 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 constantly for 2 minutes to incorporate.
04 - Gradually whisk in broth followed by milk. Bring to a simmer and cook until the sauce thickens, about 4 to 5 minutes.
05 - Add diced turkey, peas, salt, pepper, and thyme to the sauce. Remove from heat.
06 - Transfer filling to a 9-inch pie dish. Roll puff pastry to fit the dish, place atop the filling, seal the edges, trim excess, and cut slits for steam.
07 - Brush the puff pastry evenly with beaten egg for a golden finish.
08 - Bake for 30 to 35 minutes until the pastry is golden brown and crisp.
09 - Allow the dish to rest for 10 minutes before serving.