In a large skillet, heat 2 tablespoons of olive oil and 1 tablespoon of butter over medium heat. Once melted, add onions, sugar, and a pinch of salt and pepper. Cook undisturbed for 5 minutes, then stir and continue cooking for another 15 minutes until golden brown. Add balsamic vinegar and thyme, stirring for an additional 5 minutes.

Roll out 1 sheet of puff pastry on a floured surface until it’s about 1/4 inch thick. Cut into 6 equal rectangles of approximately 4x6 inches. Transfer to a parchment-lined baking sheet and preheat your oven to 400°F (200°C).

Spread caramelized onions over each pastry rectangle, leaving a half-inch border. Crumble goat cheese on top and sprinkle with additional thyme if desired.

Bake the tarts in the preheated oven for about 15 minutes, checking every 2 minutes. They should puff up and turn golden brown.

Let the tarts cool on the baking sheet for about 5 minutes before serving.