Preheat oven to 400 degrees F. In a 9 or 10-inch cast-iron skillet, melt 1 stick of the butter over medium heat. Stir in sugar and spices, bring to a slow boil and cook, stirring, until mixture is a smooth caramel-colored sauce. Fit in apple quarters, round-side down; reduce heat to low.

On a lightly-floured work surface roll out puff pastry until it is 1-inch in diameter larger than skillet. Prick all over with a pastry docker or with tines of a fork. Cut the remaining 1/2 stick butter into small pieces and dot the tops of the apples. Fit pastry round carefully over apples and butter, tucking in edges. Set the skillet on a rimmed baking sheet and transfer to the oven and bake 25 minutes, or until pastry is golden brown.

Remove skillet from oven. To unmold, loosen pastry around edges with a small, sharp knife.

Invert a serving plate over top of pan. Using oven mitts, hold both pan and plate and in one motion, flip pan upside-down and set down. Tap bottom of pan gently to make sure tart is released. Lift off pan and serve warm or at room temperature.