In a large skillet, heat 2 tablespoons of olive oil and 1 tablespoon of butter over medium heat. Add 2 thinly sliced onions, 1 teaspoon of sugar, and a pinch of salt and pepper. Cook undisturbed for about 5 minutes until the onions turn translucent. After 15 minutes, stir occasionally until golden and caramelized, then add 1 tablespoon of balsamic vinegar and 1 teaspoon of fresh thyme. Cook for an additional 5 minutes.

While the onions are caramelizing, preheat your oven to 400°F (200°C). On a lightly floured surface, roll out 1 sheet of puff pastry into a rectangle about 1/8 inch thick. Use a sharp knife to cut it into 6 smaller rectangles. Arrange the puff pastry pieces on a baking sheet lined with parchment paper.

Evenly distribute the caramelized onion mixture over the puff pastry rectangles, leaving a 1/2-inch border around the edges. Crumble about 4 ounces of goat cheese over the onions.

Transfer the baking sheet to the preheated oven and bake for 15 to 20 minutes, or until the pastry is puffed and golden brown.

Once the tarts are golden and flaky, remove them from the oven and let them cool on the baking sheet for about 5 minutes. Serve warm or at room temperature.