Preheat the oven to 400 degrees. Line a baking sheet with parchment paper and set aside. Slice the onion very thin with a knife or a mandolin. Cut all of the slices in half. Put the olive oil in a skillet, add the onions, the balsamic vinegar and brown sugar. Cook until the onions are tender. Stir often so that they do not burn. While your onions are cooking, roll out the sheet of puff pastry on a piece of parchment paper. Cut out 9 circles with a 3 inch round cookie or biscuit cutter (a glass will also suffice). Set the circles of pastry on the baking sheet. Pinch the edges of each of the circles of pastry to create a small edge to the crust. Brush each the edges of each pastry circle with the egg wash. Divide the goat cheese into 9 equal portions. When the onions are tender, remove them from the heat and divide them among the 9 circles of pastry and top them with the goat cheese. Bake for 20 minutes at 400 degrees or until the crusts are golden brown. Sprinkle thyme leaves on the tarts. Recipe from Celebrate & Decorate.