In a large skillet, heat olive oil and butter over medium heat. Add sliced onions, sugar, salt, and pepper. Cook undisturbed for 15 minutes, stirring occasionally, until soft and golden. Add balsamic vinegar and thyme, cook for an additional 5 minutes until jammy.

Preheat oven to 400°F (200°C). Roll out puff pastry to 1/8-inch thick. Cut into desired shapes and transfer to a parchment-lined baking sheet. Prick the center with a fork.

Spoon caramelized onions onto the center of each pastry shape, leaving a small border.

Crumble goat cheese over the caramelized onions. Optionally sprinkle with fresh thyme leaves.

Bake for 15 minutes, or until golden brown and flaky. Watch closely to prevent burning.

Let cool on a wire rack for a few minutes before serving warm as an elegant appetizer.