Preheat oven to 350 degrees F. In a 9 or 10-inch cast-iron skillet, melt 1 1/2 sticks 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. Dot tops of apples with remaining 1/2 stick butter. Fit pastry round carefully over apples, tucking in edges. Transfer to oven and bake 25 minutes, or until pastry in 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.
Yield: 8 to 10 servings