Cut thin, ¼ inch slices off tops and bottoms of apples, then peel, quarter, and core them. Set aside.

In a large pot over medium heat, combine sugar and 3 tablespoons of water. Stir occasionally with a rubber spatula to ensure even caramelization of sugar. Cook until sugar is dissolved and light golden brown, 5 to 8 minutes. Add butter and stir until melted and caramel looks creamy, about 1 minute. Add apples to pot, and stir to coat in caramel. Caramel may clump around apples, but continue stirring often until caramel melts again, apples release some of their juices, and very little caramel remains in the bottom of the pot, about 10 minutes. Remove from heat, add apple butter, and stir to evenly coat the apples. Set aside to let apples cool slightly.

Preheat oven to 400°F.

Place puff pastry on a large piece of parchment paper and use a rolling pin to roll out dough until it is an 11 inch square. Cut puff pastry into an 11 inch circle, dock all over with a fork, and place in the refrigerator to chill.

Arrange caramelized apples in concentric circles in a 10 inch cake pan, starting with the outer ring and moving to the center. Wedge apples tightly to fit as many as you can. Remove puff pastry from the refrigerator and place it on top of the apples. Use clean hands and a large spoon to tuck edges of pastry down and around the filling, between the apples and the pan.

Place in oven and bake for 30-40 minutes, or until pastry is puffed and golden. Remove from oven and cool on a wire rack for 30 minutes.

Once cooled, run a knife around the edge of the pan to loosen any sticky caramel. Place a large plate or platter on top of the tart, and, carefully, in one smooth motion, invert tarte onto the plate. Remove pan from tarte, and replace or adjust any apples that may have moved or stuck to the pan during the inversion process.

Serve tarte tatin immediately while warm, with vanilla ice cream if desired.