Heat olive oil and butter in a skillet over medium heat. Add onions, sugar, salt, and black pepper. Cook for 15-20 minutes, stirring occasionally, until golden brown. In the last 5 minutes, add balsamic vinegar and thyme.

Preheat oven to 400°F (200°C). Roll out puff pastry to 1/8 inch thick. Cut into 6 rectangles and transfer to a lined baking sheet. Prick center of each rectangle with a fork.

Spoon caramelized onions onto each pastry rectangle, leaving a small border. Crumble goat cheese over the onions. Optionally sprinkle additional thyme on top.

Bake tarts for 15-20 minutes until the pastry is puffed and golden brown. Cool slightly before serving.