Melt butter in a large skillet over medium heat. Drain water from apples, and place them into the hot skillet. Cook and stir for about 2 minutes. Add brown sugar, and cinnamon, and cook, stirring, for 2 more minutes. Stir together cornstarch and 1 tablespoon water. Pour into the skillet, and mix well. Cook for another minute, or until sauce has thickened. Remove from heat to cool slightly.
Preheat the oven to 400 degrees F (200 degrees C).
Unfold puff pastry sheets, and repair any cracks by pressing them back together. Trim each sheet into a square. Then cut each larger square into 4 smaller squares. Spoon apples onto the center of each squares. Fold over from corner to corner into a triangle shape, and press edges together to seal. Place turnovers on a baking sheet, leaving about 1 inch between them.