Heat olive oil and butter in a large skillet over medium heat. Add the sliced onions, stirring to coat. Cook for 25-30 minutes until golden brown, stirring occasionally. In the last 5 minutes, stir in sugar, salt, pepper, and thyme. Set aside to cool slightly.

Preheat your oven to 375°F (190°C). Roll out the puff pastry on a floured surface and cut into small squares or circles. Place them on a baking sheet lined with parchment paper. Brush the edges with beaten egg.

Spread a thin layer of the caramelized onions on each puff pastry square or tart shell, leaving a small border. Sprinkle the crumbled goat cheese evenly over the onions. For a richer filling, drizzle heavy cream over the cheese.

Place the tarts in the preheated oven and bake for 15-20 minutes, or until the pastry is puffed and golden brown, and the cheese is slightly melted. Let cool for a few minutes, then garnish with fresh thyme leaves or a drizzle of honey.