Melt butter in a medium saucepan over medium heat. Add the diced apples, granulated sugar, cinnamon, nutmeg, and lemon juice. Cook, stirring occasionally, for about 5-7 minutes, or until the apples are slightly softened but still hold their shape. Stir in the flour and cook for another minute until the mixture thickens slightly. Remove from heat and let cool completely.
Preheat your oven to 400°F (200°C). Line a baking sheet with parchment paper. Gently unfold the thawed puff pastry sheet onto a lightly floured surface. Cut the pastry into 6-8 equal rectangles.
Spoon about 1-2 tablespoons of the cooled apple filling onto one half of each pastry rectangle, leaving a small border. Fold the other half of the pastry over the filling to create a turnover shape. Press the edges together with your fingers to seal, then crimp the edges with a fork to ensure they are well sealed.
Place the assembled turnovers onto the prepared baking sheet. Brush the tops with the beaten egg wash. Sprinkle generously with granulated sugar.
Bake for 20-25 minutes, or until the turnovers are puffed up and golden brown. Let cool slightly before serving.