Heat the oven to 400 degrees F. Stir the cinnamon and sugar in a shallow bowl. Beat the egg and water in a small bowl with a fork or whisk. Using a melon baller, scoop the cores out of the apples from the top, making sure to leave the bottoms intact.
Coat the apples with the cinnamon mixture. Spoon 2 tablespoons caramel sauce into each apple.
Unfold the pastry sheet on a lightly floured surface. Roll the pastry sheet into a 10x9-inch rectangle. Cut into 9 (1x10-inch) strips. Press the ends of 2 pastry strips together to make 1 (1x20-inch) strip. Repeat with 6 of the remaining pastry strips. Cut 8 leaf shapes from the last pastry strip.
Starting at the bottom, wind 1 pastry strip around 1 apple, slightly overlapping the edges of the pastry. Repeat with the remaining pastry strips and apples. Top each with 2 pastry leaves. Brush the pastries with the egg mixture. Place the pastries onto a baking sheet.
Bake for 25 minutes or until the pastries are golden brown. Serve the pastries with the ice cream.