In a large skillet, heat 1 tablespoon of olive oil and 1 tablespoon of butter over medium heat. Add 3 sliced onions, along with 1 teaspoon of sugar, and sprinkle with salt and pepper to taste. Cook undisturbed for 5 minutes, then stir occasionally for about 15-20 minutes until the onions are golden brown and jammy. Finally, stir in 1 tablespoon of balsamic vinegar and a few sprigs of fresh thyme, cooking for another 5 minutes.

While the onions are caramelizing, preheat your oven to 400°F (200°C). Roll out one sheet of thawed puff pastry on a lightly floured surface and cut it into rectangles, leaving a small border around each piece. Remove the pan from the heat, allowing the caramelized onion mixture to cool slightly before spreading it evenly over the pastry rectangles. Crumble goat cheese on top of the onions, ensuring an even distribution.

Transfer the prepared tarts onto a baking sheet lined with parchment paper. Place the baking sheet in the preheated oven and bake for 15 minutes, or until the pastry is puffed and golden brown, with the cheese melting beautifully. Keep a close eye on the tarts, checking every 2 minutes after the initial time to prevent burning.

Once baked, allow the Caramelized Onion and Goat Cheese Tarts to cool slightly on the baking sheet for about 5 minutes. This helps the flavors settle and makes handling easier. Serve warm, either on their own or paired with a glass of wine for a delightful appetizer experience.