Melt butter in a large skillet over medium-low heat. Add onions, brown sugar, and vinegar; cook, stirring frequently, until onions are golden brown and caramelized, about 15 minutes. Transfer to a bowl with a slotted spoon and cool to room temperature.
Roll puff pastry out into a 11×18-inch rectangle on a lightly floured surface. Spread Brie cheese over pastry. Layer with caramelized onions and sprinkle with caraway seeds. Roll up one long edge towards the center. Roll up from the other side so the 2 rolls meet in the center. Cut into 1/2-inch slices using a serrated knife.
Line a baking sheet with parchment paper. Arrange slices on top; flatten to 1/4-inch thickness. Refrigerate for 15 minutes.
Preheat the oven to 375 degrees F (190 degrees C).
Mix egg and water together in a small bowl to make egg wash. Brush egg wash over slices.
Bake pinwheels in the preheated oven until puffed up and golden brown, about 12 minutes.
Use cider vinegar instead of white wine vinegar if desired.
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