Heat oven to 375°F and set a rack in the center.
Melt butter in a 10-inch cast-iron skillet over medium heat; spoon 2 tbsp into a large bowl.
Stir brown sugar into the butter left in the skillet. Cook, stirring, until glossy and bubbling, 2–3 minutes. Remove from heat.
Toss apples with the reserved melted butter, granulated sugar, flour, cinnamon, nutmeg, lemon juice, vanilla, and salt until coated.
Lay one pie crust over the caramel in the skillet, gently pressing up the sides without tearing.
Pile the apple mixture into the lined skillet, mounding slightly in the center.
Cover with the second crust, trim excess, and crimp the edges to seal. Cut 4–5 steam slits on top.
Brush the top with milk and sprinkle with coarse sugar.
Bake 45–50 minutes until the crust is deep golden and the filling bubbles at the edges. Tent with foil if browning too fast.
Cool 30–45 minutes so the juices thicken. Slice and serve warm.