Preheat oven to 400 degrees F (200 degrees C). Butter 2 small ramekins (6 to 8 ounce) and dust with white sugar.
Place apple slices on a plate, overlapping slightly if necessary. Microwave on High about 45 seconds just until slices have softened a bit. Cover plate with plastic wrap and a kitchen towel.
Mix together sugar and cinnamon in a bowl.
Roll puff pastry sheet to less than 1/8-inch thickness. Using a pizza cutter, cut two 3-inch by 12-inch rectangles. Reserve remaining pieces for another use.
Spread melted butter over dough; top with a generous amount of cinnamon-sugar. Place apple slices along 1 long edge of dough, about 1/4 inch beyond edge of dough, overlapping slices slightly. Fold bottom half of dough over the apple slices to form a long "folder" of dough with rounded edges of apple slices exposed.
Beat egg and water together in a bowl. Paint surface of dough with egg wash. Sprinkle with more cinnamon-sugar, to taste.
Starting from one end, roll dough not too tightly to form a rose-shaped pastry. Seal roll with end of dough strip.
Transfer roses to prepared ramekins. Sprinkle with a little more cinnamon-sugar. Place ramekins directly on middle rack of preheated oven.
Bake until well browned, about 45 minutes. Remove ramekins with tongs and place on baking sheet to cool 5 to 10 minutes. Remove apple roses from ramekins and finish cooling on rack. Dust with confectioners' sugar before serving.