In a large skillet, heat 2 tablespoons of olive oil and 1 tablespoon of butter over medium heat. Add 3 sliced onions, 1 teaspoon of sugar, and a pinch of salt and pepper. Stir occasionally and cook for 15-20 minutes, watching for the onions to turn a rich, golden color and become tender. Finally, stir in 1 tablespoon of balsamic vinegar and a few sprigs of fresh thyme, cooking for an additional 5 minutes until the mixture is jammy.

While the onions are caramelizing, preheat your oven to 400°F. On a floured surface, roll out 1 sheet of puff pastry into a thin rectangle, then cut it into 6 equal rectangles. Leave a small border around the edges for puffing. Place the pastry pieces on a parchment-lined baking sheet. Spoon the caramelized onion mixture evenly across the tarts, leaving the borders clear.

Sprinkle about 4 ounces of crumbled goat cheese over the caramelized onions on each tart, followed by a sprinkle of extra thyme for flavor. Bake in the preheated oven for 15-20 minutes, or until the pastry is puffed and golden brown, and the cheese is melted. Keep an eye on them to ensure they don't burn. Let the tarts cool slightly before serving for the best texture and flavor.