Special equipment: 10-inch cast-iron pan
Lightly flour a work surface. Roll the puff pastry out so it is slightly larger than the top of a 10-inch cast-iron pan. Turn the pan over on top of the pastry and cut around the outside edge of the pan to remove excess. Place the puff pastry round onto a baking sheet and refrigerate until ready to use.
Add the sugar, apple cider vinegar and 1 tablespoon water to the cast-iron pan. Turn to medium-high heat and stir the ingredients together so that the sugar is completely wet. Bring the mixture to a boil and cook until the sugar turns a deep amber, 5 to 6 minutes. Remove from the heat and add the butter and sea salt flakes; the mixture will bubble up slightly. Stir the caramel until the butter is completely melted and evenly incorporated. Let cool.
Preheat the oven to 400 degrees F.
Peel the apples, then cut each into thirds around the core, leaving you three rounded discs per apple. At this point, the caramel will have cooled and perhaps even hardened; that's ok. Place the apple pieces rounded-side down into the pan on top of the caramel, overlapping the edges slightly so you have an outside circle of apples and an inside circle of apples. Turn the pan back up to medium-high heat and bring the mixture to a boil. Once at a boil, turn the heat to medium low and simmer until the apples have softened and the excess juice from the apples has evaporated, about 15 minutes. Remove from the heat.
Place the puff pastry round on top of the apples. Use a mini offset spatula or butter knife to tuck the edges of the puff pastry into the pan towards the apples, all along the edge. Whisk the egg together with a splash of water in a small bowl and then brush the puff pastry with the egg wash. Bake until the puff pastry is a deep golden brown, about 12 minutes. Let cool 15 minutes.
Using a serving platter that is at least 2 inches wider than the pan and has a small lip on it to catch excess caramel, place it over the top of the cooled tarte. Place a kitchen towel on top of the platter and then fold it down over the cast iron. Get a firm grip of the platter and cast iron in both hands before flipping the cast iron over, away from yourself. Remove the cast iron.
Serve the tarte tatin immediately with whipped cream or ice cream if desired.