Heat the oven to 425°. In a small bowl, combine the sugar and cinnamon. Set aside.
On a floured surface, roll out one of the puff-pastry sheets to a 13-by-13-inch square. Cut the square in half and cut each half into a 9-by-6-inch rectangle. If you want to make the optional decoration in Step 3, reserve the scraps. With the beaten egg, lightly brush 1 inch along the edges of one of the rectangles. Put one of the apples, upside down, in the middle of the rectangle and pour enough of the cinnamon sugar into the center cavity to reach the top. Gather the four corners of the pastry and pinch them together. Pinch the edges together to seal. With scissors, trim any excess pastry from where the edges meet. Repeat with the remaining pastry, apples, and cinnamon sugar.
If you like, use a small knife to cut eight small leaves from the scraps of pastry. Brush the backs of the leaves with the beaten egg and attach two of them to the top of each dumpling.
Chill the dumplings for 15 minutes. Put them on a heavy baking sheet and lightly brush them with the beaten egg. Bake the dumplings for 8 minutes. Reduce the heat to 400° and continue cooking until golden, 20 to 22 minutes longer. Don't bake too long or the dumplings may burst.