Preheat oven to 400 degrees F (200 degrees C). Butter a 9×13 inch pan.
On a lightly floured surface, roll pastry into a large rectangle, about 24 by 16 inches. Cut into 6 square pieces. Place an apple on each pastry square with the cored opening facing upward. Cut butter into 8 pieces. Place 1 piece of butter in the opening of each apple; reserve remaining butter for sauce. Divide brown sugar between apples, poking some inside each cored opening and the rest around the base of each apple. Sprinkle cinnamon and nutmeg over the apples.
With slightly wet fingertips, bring one corner of pastry square up to the top of the apple, then bring the opposite corner to the top and press together. Bring up the two remaining corners, and seal. Slightly pinch the dough at the sides to completely seal in the apple. Repeat with the remaining apples. Place in prepared baking dish.
In a saucepan, combine water, white sugar, vanilla extract and reserved butter. Place over medium heat, and bring to a boil in a large saucepan. Boil for 5 minutes, or until sugar is dissolved. Carefully pour over dumplings.
Bake in preheated oven for 50 to 60 minutes. Place each apple dumpling in a dessert bowl, and spoon some sauce over the top.