Peel, core, and slice apples into thin wedges. Melt butter in a skillet over medium heat, then add apples, both sugars, cinnamon, nutmeg, and lemon juice.
Cook 5–7 minutes until softened. If runny, stir in cornstarch slurry and cook 1–2 more minutes until thickened. Let cool slightly.
Preheat oven to 400°F (200°C). Roll out thawed puff pastry on a lightly floured surface and cut into 8 rectangles or squares. Place half on a parchment-lined baking sheet.
Spoon apple filling into the center, drizzle with caramel sauce, and top with another pastry piece or fold over into triangles. Seal edges with a fork and brush tops with egg wash.
Bake 18–22 minutes until puffed and golden. Cool slightly, drizzle with more caramel sauce, and dust with powdered sugar. Serve warm with whipped cream or vanilla ice cream.