In a large skillet cook your onions in the butter, brown sugar, and vinegar over medium low heat until the onions are golden brown, stirring frequently.
Remove the onions with a slotted spoon; cool to room temperature.
On a lightly floured surface, roll your puff pastry into an 11-inch x 8-inch rectangle.
Spread the softened Brie cheese over the pastry.
Cover the cheese with the onions; sprinkle with the caraway seeds.
Roll up one long side to the middle, and the other side to the middle, so you have two rolls. Using a serrated knife, cut the rolls into 1/2-inch slices.
Place them on a parchment-paper-lined baking sheet and flatten to about a 1/4-inch thickness.
Put in the refrigerator for about 15 minutes.
Preheat your oven to 375 degrees.
In a small bowl beat an egg and some water; remove your onion Brie palmiers from the refrigerator and brush the egg mixture over the tops.
Bake for 12 to 14 minutes until puffed and golden brown.
Serve warm and enjoy!