In a large skillet, heat 2 tablespoons of olive oil and 1 tablespoon of butter over medium heat until bubbling. Add 3 sliced onions, 1 tablespoon of sugar, ½ teaspoon of salt, and a pinch of pepper. Cook undisturbed for 5 minutes, then stir occasionally every 5 minutes until the onions are a deep golden brown, about 15-20 minutes. Add 1 tablespoon of balsamic vinegar and the leaves from 1 teaspoon of fresh thyme, allowing them to cook together for an additional 5 minutes until the mixture is jammy.

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 ¼ inch thick. Using a sharp knife or pastry wheel, cut the dough into 6 rustic rectangles and place them on a baking sheet lined with parchment paper.

Using a spoon, carefully spread the caramelized onion mixture evenly over each pastry rectangle, leaving a small border around the edges. Crumble around 4 ounces of goat cheese over the top of the onions. Sprinkle additional thyme leaves or a drizzle of honey if desired.

Place the assembled Caramelized Onion and Goat Cheese Tarts in your preheated oven and bake for about 15 minutes, or until the pastry is puffed and golden brown. Start checking for doneness around the 12-minute mark.

Once baked, remove the tarts from the oven and allow them to cool on the baking sheet for about 5 minutes. Serve warm and enjoy.