Prepare the Apple Filling. In a skillet over medium heat, melt butter and add diced apples. Stir in brown sugar, cinnamon, nutmeg, and vanilla extract. Cook for 5-7 minutes, until the apples soften and release their juices. Mix cornstarch with a teaspoon of water, then stir into the apple mixture. Cook for another 1-2 minutes, until the filling thickens. Remove from heat and let cool.
Prepare the Puff Pastry. Preheat oven to 400°F and line a baking sheet with parchment paper. Roll out the puff pastry on a lightly floured surface and cut into 6 equal squares.
Fill and Fold the Turnovers. Place a spoonful of apple filling in the center of each puff pastry square. Fold the pastry diagonally to create a triangle. Use a fork to seal the edges and prevent filling from leaking if you prefer. Place the turnovers on the prepared baking sheet.
Brush and Bake. In a small bowl, whisk the egg and milk together for the egg wash. Brush the egg wash over the turnovers and sprinkle with coarse sugar if desired. Bake for 18-22 minutes, or until golden brown and puffed up. Let them cool slightly before serving.