Unfold puff pastry on a lightly floured surface. Roll pastry into a 14-inch square. Using a fluted pastry cutter or table knife, cut pastry into four 7-inch squares. Set aside.
Preheat oven to 400 degree F. Core apples; peel apples, if desired. If necessary, trim bottoms of apples so they stand upright. Place an apple in the center of each pastry square. In a small bowl combine sugar and cinnamon; spoon into centers of apples.
In another small bowl beat egg and the water with a fork. Moisten the edges of the pastry squares with egg mixture; fold corners to center over fruit. Pinch to seal, pleating and folding pastry along seams as necessary. Place dumplings in a 13x9x2-inch baking pan. Brush wrapped apples with egg mixture.
Bake dumplings in preheated oven about 35 minutes or until fruit is tender and pastry is brown.
Meanwhile, for sauce, in a microwave-safe 2-cup glass measure combine caramel topping and pecans. Microwave, uncovered, on 100 percent (high) power for 30 to 60 seconds or until heated through.