Wash, peel and core the apples. Slice them thinly. In a medium sauce pan, add butter, sugar and cinnamon and set on low heat until the sugar dissolves. Add the apple slices and cook until the apples are tender. Drain the apples but reserve the caramelized liquid. Set aside.
In a food processor or blender, add the pecans, lemon juice and apple sauce and pulse until the mixture turns to a paste with some little pecan pieces.
Preheat oven to 400 F. Line a baking sheet with parchment paper. Cut circles from the rolled puff pastry that are 4 and 1/2 inches in size. Brush the surface with the beaten egg using a pastry brush. Arrange the 4 circles in the baking sheet. From the other 4 circles, cut smaller circles that are 3 and 1/2 inches in size. Leave the centers and take the outer rings and position them on top of the bigger 4 circles.
Spread the pecan mixture on top of the assembled puff pastry, using the cut rings as borders. Top with the apple slices. Bake them uncovered for 15 minutes. The pastry should be lightly golden.
Recipe adapted from Best Ever Recipes by Wiley and Sons.