In a large skillet, cook the onions, butter, brown sugar and vinegar over medium-low heat until onions are golden brown, stirring frequently. Remove with a slotted spoon; cool to room temperature.
On a lightly floured surface, roll puff pastry into an 11-in. x 8-in. rectangle. Spread Brie over pastry. Cover with the onions; sprinkle with caraway seeds.
Roll up one long side to the middle of the dough; roll up the other side so the two rolls meet in the center. Using a serrated knife, cut into 1/2-in. slices. Place on parchment paper-lined baking sheets; flatten to 1/4-in. thickness. Refrigerate for 15 minutes.
In a small bowl, beat egg and water; brush over slices. Bake at 375° for 12-14 minutes or until puffed and golden brown. Serve warm. Yield: 1-1/2 dozen.