Add the sugar to a 10-inch cast-iron skillet over medium-low heat. Cook gently, swirling often until the sugar melts and becomes golden brown, 12 to 15 minutes. Remove the skillet from the heat.
Peel, core and halve the apples. Begin arranging the apple halves in a circular shape around the border of the skillet, standing them on the core ends. Fill the center part of the skillet with a smaller circle of the apples. Fill in any gaps with the smaller top and bottom pieces. Because the fruit will shrink considerably as it cooks, it is better to crowd the pan with fruit.
Place the skillet back on the stovetop on a fairly high heat and cook until the fruit starts to become tender and soak up the caramel at the bottom, 12 to 15 minutes. Remove from the heat again and rotate each apple piece halfway so the tops of the fruit are now sitting in the caramel on the bottom. Return the skillet to the heat and cook for an additional 5 to 8 minutes. Remove the skillet from the heat.
Roll the puff pastry dough on a lightly floured surface so it's larger than the skillet by 1 inch. Gently drape it over the skillet. Tuck the dough edges loosely inside the skillet so it covers the fruit (as if you are tucking the fruit into a pastry bed).
Bake until bubbly around the edges, golden brown and delicious, 25 to 30 minutes. Remove and cool 5 to 10 minutes. Then, scoop and spoon onto plates. Serve with vanilla ice cream.