Position a rack in lower third of oven and preheat to 400F. Line a baking sheet with a piece of parchment paper.
Combine apples, jam and cornstarch in a medium bowl.
Mix sugar and cinnamon in a separate small bowl.
On a lightly floured surface, roll out puff pastry into an 8-by-12-inch rectangle.
Cut pastry into 6 4-inch squares with a sharp knife or pizza cutter.
Spoon some apple mixture into center of each square, leaving a 1-inch border.
Brush borders with beaten egg. Fold each square into a triangle and press edges with tines of a fork to seal (be sure to seal tightly). Using a sharp paring knife, cut 2 small steam vents in top of each turnover.
