In a large skillet, heat olive oil and butter over medium heat until melted and shimmering. Add sliced onions, sugar, and salt and pepper. Cook undisturbed for about 15 minutes, stirring every 5 minutes until onions turn a deep golden brown and begin to caramelize beautifully.

Once the onions are caramelized, drizzle in balsamic vinegar and sprinkle with fresh thyme. Continue to cook for an additional 5 minutes, stirring occasionally, until the onions are jammy and richly flavored. Remove from heat and let cool slightly.

Preheat your oven to 400°F (200°C). On a lightly floured surface, roll out the thawed puff pastry into a rectangle and cut it into 6 equal rectangles.

Place the puff pastry rectangles on a baking sheet lined with parchment paper. Spoon the cooled caramelized onion mixture onto each rectangle, spreading it evenly and leaving a small border. Crumble goat cheese on top of the onions and sprinkle a little extra thyme.

Bake the assembled tarts for 15 minutes, or until the pastry is golden brown and the cheese is bubbly. Check every 2 minutes to avoid overbrowning.

Once baked, cool the tarts for a few minutes on the baking sheet. Serve warm to enjoy the delightful combination of sweet caramelized onions and tangy goat cheese.