Heat equal parts olive oil and butter in a large skillet over medium heat. Add thinly sliced onions, a pinch of sugar, salt, and pepper. Stir gently and cook undisturbed for about 5 minutes.

Reduce heat to medium-low and stir the onions occasionally. Cook for about 15 minutes until they develop a rich golden color and become tender.

Add balsamic vinegar and thyme to the skillet. Stir and cook for an additional 5 minutes allowing the vinegar to reduce.

Preheat your oven to 400°F (200°C). Roll out the puff pastry on a lightly floured surface and cut it into rectangles or circles about 3 inches in size.

Place the cut pastry on a baking sheet lined with parchment paper. Spread a generous spoonful of the caramelized onion mixture onto each pastry.

Crumble goat cheese over each tart and sprinkle with fresh thyme.

Bake for about 15 minutes, checking every 2 minutes for golden brown pastry and melting cheese.

Remove from the oven and let cool for a few minutes before serving warm.