In a large skillet, heat olive oil and butter over medium heat until melted. Add onions, sugar, salt, and pepper. Cook undisturbed for 5 minutes, then stir and continue cooking until deeply golden and caramelized, about 15–20 minutes. Mix in balsamic vinegar and thyme, cooking for an additional 5 minutes.

Preheat oven to 400°F (200°C). Roll out puff pastry on floured surface to about 1/4-inch thickness. Cut into rectangles, about 4x6 inches. Transfer to a baking sheet lined with parchment paper, spreading caramelized onion mixture evenly over each.

Crumble goat cheese over the caramelized onion layer and optionally sprinkle with extra thyme leaves. Chill in the fridge for 10 minutes.

Bake for 15–20 minutes, or until pastry is puffed and golden brown, and cheese is melted. Monitor closely for an ideal crispy crust.

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