At least one day before you plan to cook the tart, prepare the apples: Slice off the bottom of each apple so it has a flat base. Peel and quarter the apples. Use a small sharp knife to trim the hard cores and seeds from the center of each quarter; don’t worry about being too neat. Transfer to a bowl and refrigerate, lightly covered, for at least one day or up to three days. (This key step reduces the amount of liquid in the tart. Don’t worry if the apples turn brown; they will be browned during the cooking anyway.)
When ready to cook, heat oven to 375 degrees (or 350 if using convection). Thickly coat the bottom of a 10-inch heavy ovenproof skillet, preferably nonstick metal, with butter. Sprinkle sugar evenly on top.
Cut one piece of apple into a thick round disk and place in the center of the skillet to serve as the “button.” Arrange the remaining apple pieces, each one standing on its flat end, in concentric circles around the button. Keep the pieces close together so that they support one another, standing upright. They will look like the petals of a flower.
On a floured surface, roll out the puff pastry* about 1/8-inch thick. Place an upside-down bowl or pan on the pastry and use the tip of a sharp knife to cut out a circle about the same size as the top of your skillet. Lift out the circle and drape gently over the apples. Use your hands to tuck the pastry around the apple pieces, hugging them together firmly.
Place the skillet on the stovetop over medium heat until golden-brown juice begins to bubble around the edges, 3 minutes (if the juices keep rising, spoon out as needed to remain level with pastry). Keep cooking until the juices are turning darker brown and smell caramelized, no longer than 10 minutes more.
Transfer skillet to the oven and bake 45 to 50 minutes, until puff pastry is browned and firm.
Let cool 5 minutes, then carefully turn out onto a round serving plate. (Or, if not serving immediately, let cool completely in the pan; when ready to serve, rewarm for 15 minutes in a 350-degree oven before turning out.) If any apples remain stuck in the pan, gently use your fingers or a spatula to retrieve them, and rearrange on the pastry shell. Cut in wedges and serve warm with heavy cream, crème fraîche or vanilla ice cream.

*Puff pastry recipe: http://allrecipes.com/recipe/159432/puff-pastry/
Mix the flour and salt together in a large bowl or in the bowl of a stand mixer that is fitted with a dough hook. Gradually stir in water until the dough holds together enough to clean the sides of the bowl. You may not need the full amount of water. Shape into a flat ball, and allow to rest for at least 10 minutes.
Place the butter between two pieces of plastic wrap and pound into a flat disc using a rolling pin or other heavy object. Refrigerate until firm, about 20 minutes.
On a lightly floured surface, roll out the dough into a large rectangle about 1/2 inch thick. Place the disc of chilled butter in the center and fold the two ends over it so that it is completely encased in dough. Roll out the dough again, taking care not to let the butter break through the dough, to about 1/2 inch thickness. Fold into thirds. This is the first ''turn''. Rotate the dough 90 degrees and roll out into a rectangle again. Fold into thirds. By this time the butter is starting to warm up. Place the dough on a baking sheet and mark it with two pokes from your finger (two turns). Wrap in plastic and refrigerate for at least 30 minutes.
Repeat this rolling, folding and turning two more times, then refrigerate until firm. Repeat two more times for a total of 6 ''turns''. Wrap and refrigerate. The dough is now ready to roll out and use in any recipe calling for puff pastry. Roll the dough out as thin as 1/4 inch to make pastries. Bake in a preheated oven of at least 400 degrees F (200 degrees C) to get the maximum puff from your pastry.