Heat the oven to 400ºF.
Melt 3 tablespoons unsalted butter in a 10-inch cast iron or oven-safe straight-sided skillet over medium-high heat. Add ½ diced medium yellow onion, 2 small sliced carrots, 2 sliced small celery stalks, and 1 ½ teaspoons fresh thyme leaves. Cook, stirring occasionally, until the onion is tender, 5 to 6 minutes.
Sprinkle ¼ cup plus 2 tablespoons all-purpose flour over the vegetables. Cook, stirring constantly, for 1 minute to cook the floury taste out. Stir in 1 ½ cups low-sodium chicken broth, 1 cup plus 2 tablespoons whole milk, 1 teaspoon kosher salt, and ¼ teaspoon black pepper. Cook, stirring frequently with a wooden spoon, until it comes to a simmer and starts to thicken. Let simmer for 2 minutes more.
Remove the pan from the heat. Stir in 12 ounces cooked turkey and ½ cup frozen peas. Taste and season with more kosher salt and black pepper as needed.
If needed, roll out 1 pie dough into a round about 10 inches in diameter. Place the pie dough over the turkey filling. Brush the dough with a thin layer of milk. Use a sharp knife to cut 4 (1-inch) slits in the top of the crust to vent.
Bake until the filling is bubbling and the crust is golden-brown, 20 to 25 minutes. Let cool for 5 minutes before serving.
