Defrost the puff pastry according to the package directions.
Core and slice apples into very thin slices, around 1/8" thick or less if possible.
Place the water in a medium saucepan, and bring it to a simmer over medium heat. Add the apple slices and simmer them for 3 minutes. You just want to cook them slightly so they're pliable and can easily bend. Do not let them get overly soft or mushy.
Strain the apples out of the water and place them on a baking sheet covered with a paper towel to drain. Let them cool to room temperature.
Preheat the oven to 425 F. and spray a muffin pan with nonstick cooking spray. Flour your work surface and rolling pin. Roll out one sheet of puff pastry to a 12x12-inch square. Cut the square into six long strips.
Slice the cheese into paper-thin slices. Give the cranberry sauce a few good stirs to break up any clumps.
Brush the bottom of one strip of puff pastry dough with the beaten egg. Start laying apple slices along the top of the puff pastry strip, with the rounded part sticking above the dough. Overlap the slices slightly as you lay them down, and leave a small margin uncovered at the end of the strip.
Spread a little cranberry sauce on the bottom edge of the apples. Don't use too much or it will make your rose look messy as it bakes.
Add the cheese slices on top of the apples, and roll the puff pastry up. Brush a little egg onto the uncovered margin at the end of the strip, then press the pastry roll closed. Pinch the excess dough at the bottom of the pastry together.
Place the puff pastry rose in the prepared muffin tin, and repeat until all of the roses are made.
Bake at 425 F for 25-29 minutes, rotating halfway through. If the apple slices start to get too dark or burn along the edges, cover the pan loosely with foil. They're done when they're puffed and they look golden brown along the sides.