In a skillet heat apples, butter, brown sugar, cinnamon, lemon juice and cornstarch until boiling, stirring frequently. Simmer for about 5 minutes until it thickens slightly. Let it cool slightly.
Preheat oven to 400º F.
Unfold puff pastry on a lightly floured surface. Gently roll puff pastry into a 12 inch by 12 inch square. Using a sharp knife or pizza cutter, divide into 4 equal squares.
Scoop about 1/4 cup of apple mixture into the corner of each square of pastry. Fold the pastry diagonally over the apple filling and seal by using a fork to press the edges together. Repeat until you have 8 turnovers. Place on a lined baking sheet, about 2 inches apart.
Mix together egg and water until well blended. Brush the egg wash over all of the pastry triangles.
Sprinkle with sugar if using. Bake for 18-22 minutes until puffed up and golden brown. Remove from oven and let cool before serving.