In a large skillet, heat 2 tablespoons of olive oil and 1 tablespoon of butter over medium heat. Add 3 thinly sliced onions, 1 tablespoon of sugar, and a pinch of salt and pepper. Cook undisturbed for 5 minutes, then stir occasionally until the onions start to soften and become translucent.

Continue cooking the onions for an additional 15 minutes, stirring occasionally. Add 1 tablespoon of balsamic vinegar and a few sprigs of fresh thyme. Cook for 5 more minutes until the onions are a deep golden brown and have a jammy texture.

Preheat your oven to 400°F (200°C). Roll out 1 sheet of puff pastry until it's about 1/8 inch thick. Cut into rectangles, roughly 3x4 inches each. Place on a parchment-lined baking sheet.

Spread a spoonful of the caramelized onion mixture evenly over each pastry rectangle, leaving a small border around the edges.

Crumble 4 ounces of goat cheese over the caramelized onions on each tart, adding extra thyme if desired.

Bake the tarts for 15 minutes or until golden brown and puffed up. Check every 2 minutes for doneness.

Let the tarts cool slightly on a wire rack for about 5 minutes before serving warm.