Preheat the oven to 400 degrees F. Melt the butter in a large skillet over medium heat. Add the onion, carrots, and celery and cook for 5 minutes until the vegetables begin to soften.
Sprinkle the flour over the vegetables and stir for 1 minute to form a roux. Slowly pour in the chicken broth while stirring to avoid lumps.
Add the milk, thyme, salt, and pepper. Simmer the mixture for 3 minutes until it thickens slightly.
Stir in the diced turkey and frozen peas. Cook for 2 more minutes, then remove the skillet from the heat.
Transfer the filling to a 9-inch baking dish. Lay the puff pastry sheet over the top and trim any excess edges. Cut a few small slits in the pastry to allow steam to escape.
Bake for 25 minutes until the crust is golden brown and the filling is bubbling. Let the pie rest for 5 minutes before serving.