Thaw the puff pastry at room temperature for 30-40 minutes, until it is easy to unfold.
While the puff pastry is thawing, make the apple filling. Heat the butter in a medium pot over medium heat. Add the apples, sugar, cinnamon, and lemon juice, and bring the mixture to a simmer. Mix the corn starch with 1 tablespoon of water until dissolved, then stir into the apple mixture. Reduce heat and simmer 10-15 minutes, until the filling is thickened and the apples are soft. Let cool to room temperature.
Preheat the oven to 400 and line a baking sheet with parchment or spray with cooking spray.
Roll the puff pastry out into a rectangle that is approximately 12 x 18 inches. Cut the pastry into six squares. Scoop 2 tablespoons of apple filling onto each square of pastry. Fold each pastry square in half and use a fork to seal.
In a small bowl, whisk together the egg and 1 tablespoon water. Brush the egg wash over the pastries and sprinkle with the coarse sugar. Bake 15-20 minutes, or until golden brown. Let cool 5-10 minutes before serving.