Thaw and unfold the pastry as the package directs. Preheat oven to 400 degrees F. On a lightly floured surface, roll the pastry into a 16 inch square and, using a fluted pastry cutter or a kitchen knife, cut into 4 equal squares.
Peel and core apples; trim bottoms so that apples sit level. Place one apple in the center of each pastry square. Combine brown sugar and cinnamon; spoon into apples.
Beat the egg and water together in a small bowl and moisten the edges of the pastry with the mixture. Bring the pastry up around the apples, pleating or trimming excess pastry as needed, and pinch the edges together to form a seal.
Place the dumplings in an ungreased 9 x 13 inch baking dish. Brush them with the egg mixture; sprinkle with the granulated sugar. Bake for about 35 minutes, until golden brown.
For the sauce, combine the caramel topping and pecans, if desired, in a microwave-safe dish. Microwave, uncovered, on HIGH for 30 seconds, until heated through. Spoon the sauce onto each plate and set the dumplings on top or drizzle the sauce over the top. Serve warm.