Preheat the oven to 400°F and grease two ramekins
Cut the apples in half, remove the core, but keep the red peel. Slice them as thin as possible and place on a plate. Microwave for about 45 seconds until they slightly soften and become limp but not mushy. Cover the apples with a plastic wrap and towel while prepping the pastry to ensure they remain soft.
Roll out the puff pastry and cut into 2 strips, each 1/8-inch thick, about 3” wide by 12 to 14” long. There will be extra pastry left
Spread melted butter over the surface. Mix the sugar and cinnamon together in a small bowl, and sprinkle the cinnamon sugar over the pastry dough.
Place the apple slices along the edge, starting about 1 inch in from the end, overlapping slightly, with the round part up, partly sticking out of the dough
Fold the bottom half of the dough over the apples. Mix the egg with 1 tablespoon of water to make an egg wash, then brush the egg wash on top. Sprinkle more cinnamon sugar on top.
Fold the puff pastry "closing" the apples as you roll.
Place inside the greased ramekin, and repeat for the other strips
Bake in the preheated oven for 40-45 minutes, or until the pastry is well browned.
Allow to cool and serve with vanilla ice cream and caramel sauce, if desired.