Preheat the oven to 400 degrees F. Line a baking sheet with parchment paper or a silicone nonstick baking sheet. Set aside. 

Melt the butter in a small saucepan over medium heat. Add the onions and cook, stirring occasionally, for at least 20 minutes until deeply brown and they smell sweet. 

Add the balsamic vinegar, salt, and brown sugar. Cook for another 5 minutes to reach a consistency similar to jam. Set aside to cool. 

Cut the puff pastry into squares. Use a knife, cut an L shape on the bottom and right hand side. Cut another upside down L shape on the other side of each piece. See process picture. 

Gently place the cut corner over the pastry so it rests on the opposite side. Do the same with the other side. This should have a boat-like effect where each tart now has an outer edge to hold the filling in. 

In a small bowl whisk the egg with about 1 tsp of water till frothy.

Brush each pastry square with the egg wash. 

Place a piece of brie into each tart square then add some caramelized onion on top of the cheese. 

Place the tarts on the prepared baking sheet then bake for about 20 minutes or until golden brown. 

Remove the tarts from the oven, some cheese may leak out during cooking which is fine. Enjoy!