Preheat the oven to 400 degrees Fahrenheit.
Unfold each sheet of puff pastry and lay out on a clean, dry surface. Use a sharp knife to cut each sheet into 6 strips, for a total of 12 strips of puff pastry. Set aside.
Cut each apple into quarters and remove the core with a paring knife. Use a sharp knife or a mandolin to slice the apple quarters into very thin slices (the thinner the better so the apples will be able to bend when you roll the tarts into roses).
Lay the slices of apples on the top third of each strip of puff pastry, with the red apple peels facing out. A bit of each slice of apple should be slightly off of the top of the pastry.
In a small bowl, combine the melted butter, brown sugar, flour, and cinnamon (the mixture will be thick and paste-like). Use a pastry brush to brush some of the mixture on the puff pastry right below the apple slices.
Fold the bottom half of the puff pastry up over the brown sugar mixture and the bottom half of the apple slices.
Starting at one end, roll each of the pastries into a coil, like a cinnamon roll. Place into a greased muffin tin cup. Note: If some of the apples pop out or break while you’re rolling, just keep going. You can rearrange the apples once you place the tart in the muffin cup. This process can be a bit messy.
Repeat steps 4-7 with each strip of puff pastry.
Bake at 400F for 22-25 minutes until the pastry is golden brown.
Remove the tarts from the oven and place on a cooling rack to cool for 10 minutes.
Drizzle each tart with caramel sauce.