Preheat the oven to 400 degrees F. Line two baking sheets with parchment paper.
Lightly flour your work surface. Unfold puff pastry sheets. Using a rolling pin, roll over the pastry to seal any perforations. Cut each sheet into 4 squares.
In a large bowl, stir together apples, brown sugar, cinnamon and lemon juice.
Scoop about 2 Tablespoons of the apple mixture into the center of each puff pastry square. Fold the pastry across diagonally to form triangles. Using a fork, crimp edges. Place turnovers on baking sheets, about 2inches apart. Cut two small slits in the tops of each turnover.
Whisk together egg and water in a small bowl. Brush turnovers with egg wash. Sprinkle with coarse sugar (optional).
Bake for 15 to 20 minutes, until golden brown and puffed. Transfer turnovers to a rack to cool. Serve warm or at room temperature.