How to Make It
In a very large, deep skillet, cook the sugar, water, butter and lemon juice over high heat until a pale amber caramel forms, 15 minutes. Add the apples and cook over moderate heat, turning occasionally, until very soft but not falling apart, 30 minutes. Transfer the apples to a large plate and let cool. Pour the caramel into a small bowl; let cool.
Preheat the oven to 375° and line 2 baking sheets with parchment paper. On a lightly floured surface, roll out each sheet of puff pastry to a 10-by-15-inch rectangle. Cut each sheet into six 5-inch squares. For each turnover, lightly brush the edge of a pastry square with water. Place 2 cooled apple quarters in the middle and spoon a scant 1 tablespoon of the caramel on top. Fold the pastry over to form triangles and press the edges to seal. Transfer to the baking sheets.
Brush the turnovers with water and sprinkle with coarse sugar. Place the baking sheets in the oven and lower the temperature to 325°. Bake the turnovers for about 35 minutes, until golden and puffed. Let cool before serving.