Use a 9-inch cake pan as a template, and cut a 9-inch circle from the puff pastry. Poke it all over with a fork.
In a large saucepan over medium-heat, combine the butter and sugar. Cook about 5 minutes, until light brown in color (stirring constantly). Add the apples, stirring until they are coated in a thick layer of caramel. Cook for about 15 to 20 minutes, turning the apples constantly so that they bathe in the caramel. Remove from the heat when the caramel has reduced and little remains in the bottom of the pan.
Position the apple slices in concentric circles on the bottom of the cake pan. Press the apples tightly against each other, then pour the remaining caramel over the top.
Lay the circle of puff pastry on top. Tuck the puff pastry down the sides of the pan.
Bake for 45 to 50 minutes, until golden brown. Let cool in the pan for one hour.
Invert the Tarte Tatin onto a plate. Serve slices with vanilla ice cream.