Preheat oven to 375°F (190°C). Roll out the puff pastry and cut it into six squares. In a small bowl, mix brown sugar, cinnamon, and nutmeg. Place an apple quarter in the center of each pastry square and sprinkle with the sugar mixture. Add a small cube of butter on top.
Fold the pastry around the apple, sealing the edges with a little water. Place the wrapped apples seam-side down in a greased baking dish. Brush each dumpling with beaten egg for a golden finish.
In a saucepan, combine water, granulated sugar, brown sugar, butter, and cinnamon. Bring to a gentle boil and simmer for 5 minutes, then stir in vanilla extract. Pour the syrup over the dumplings in the baking dish.
Bake for 35-40 minutes, until the pastry is golden brown and crispy. Let cool slightly before serving. Enjoy warm with vanilla ice cream or whipped cream!