Melt the butter in a medium skillet over medium heat. Add the diced apples, brown sugar, cinnamon, nutmeg, and lemon juice. Cook the mixture for about 5 to 7 minutes, stirring occasionally, until the apples are tender and the liquid has thickened into a light syrup. Remove from heat and let the filling cool completely before using. Cooling is essential to prevent the pastry from becoming soggy.

Preheat your oven to 400°F (200°C) and line two baking sheets with parchment paper. On a lightly floured surface, gently unfold the thawed puff pastry sheets. Use a rolling pin to lightly smooth out any creases. Cut each sheet into four equal squares, giving you a total of eight squares.

Place about 2 tablespoons of the cooled apple mixture into the center of each square. Avoid overfilling, as this will make sealing difficult. Whisk the egg and water together in a small bowl. Lightly brush the edges of the pastry square with the egg wash. Fold one corner over the filling to the opposite corner, creating a triangle. Press the edges firmly with a fork to crimp and seal them shut.

Place the turnovers on the prepared baking sheets. Brush the tops of each turnover generously with the remaining egg wash. Use a sharp knife to cut two small slits in the top of each turnover to allow steam to escape. Bake for 18 to 22 minutes, or until the pastry is puffed and a deep golden brown.

Move the baked turnovers to a wire cooling rack. While they are still warm, whisk together the powdered sugar and milk in a small bowl until smooth. Drizzle the glaze over the turnovers using a spoon or a whisk. Serve warm for the best texture and flavor.