Prepare your workstation by preheating your oven to 400°F (200°C) and lining a baking sheet with parchment paper. Keep your thawed puff pastry in the refrigerator until you’re ready to use it.
Make the apple filling by combining diced apples, both sugars, lemon juice, cinnamon, nutmeg, and salt in a medium saucepan over medium heat. Cook for about 5-7 minutes, stirring occasionally, until apples begin to soften but still maintain their shape.
Add the butter to the apple mixture and stir until melted. In a small bowl, mix cornstarch and water to create a slurry, then pour into the apple mixture and cook for another 1-2 minutes until the filling thickens. Remove from heat and allow to cool completely.
Roll out the puff pastry on a lightly floured surface to smooth out any creases. Cut each sheet into 4 equal squares, giving you 8 squares total.
Spoon the cooled filling onto one half of each pastry square, leaving a 1/2-inch border around the edges. Be careful not to overfill—about 2 tablespoons of filling per turnover is perfect.
Create the egg wash by beating the egg with 1 tablespoon of milk or water in a small bowl. Brush the borders of the pastry with this egg wash.
Fold the pastry over to create triangles, gently pressing the edges with your fingers first, then use a fork to crimp and seal the edges completely.
Transfer the turnovers to your prepared baking sheet, spacing them at least 2 inches apart. Use a sharp knife to cut 2-3 small slits in the top of each turnover to allow steam to escape.
Brush the tops with the remaining egg wash and sprinkle generously with coarse sugar for that bakery-style finish and extra crunch.
Bake for 18-22 minutes until the turnovers are puffed and golden brown. Some apple filling may bubble through the vents, which is perfectly normal.
Cool the turnovers on the baking sheet for 5 minutes, then transfer to a wire rack to cool for at least 10 more minutes before serving.