Heat 2 tablespoons of olive oil and 2 tablespoons of butter in a large skillet over medium heat. Add 3 sliced yellow onions, 1 teaspoon of sugar, and a pinch of salt and pepper. Cook undisturbed for about 15 minutes until deep golden brown. Stir in 1 tablespoon of balsamic vinegar and 1 teaspoon of fresh thyme, cooking for an additional 5 minutes.

Preheat your oven to 400°F (200°C). Roll out 1 sheet of puff pastry until about 1/8 inch thick. Cut into rectangles, approximately 4x6 inches each, and transfer to a baking sheet lined with parchment paper.

Spread a generous tablespoon of the caramelized onion mixture onto the center of each puff pastry rectangle, leaving a small border. Crumble 4 ounces of goat cheese over the onions and sprinkle a few thyme leaves on top.

Bake the tarts for 15-20 minutes, checking until they're golden brown and puffed.

Let the tarts cool for about 5 minutes on a wire rack before serving warm. Garnish with additional thyme or balsamic reduction if desired.