Follow these steps to create your own rich caramelized onion tarts. Each step is crucial for achieving that perfect balance of flavors:

Preparing the Caramelized Onions

1. Heat Butter and Oil: In a large skillet, melt the butter and add olive oil over medium heat.
2. Add Onions: Introduce the sliced onions to the skillet, stirring to coat them evenly in the butter and oil.
3. Cook Slowly: Reduce the heat to low and cook the onions for about 30-40 minutes, stirring occasionally. The goal is to achieve a deep brown caramelization without burning.
4. Sweeten (Optional): If you prefer sweeter onions, sprinkle sugar over them halfway through the cooking process and mix well.
5. Add Vinegar: Stir in the balsamic vinegar and season with salt and pepper. Cook for an additional 5 minutes, then remove from heat and let cool.

Assembling the Tarts

6. Preheat Oven: Heat your oven to 400°F (200°C).
7. Prepare Puff Pastry: On a lightly floured surface, roll out the puff pastry sheet to a thickness of about 1/8 inch. Cut into desired shapes (squares or circles work well).
8. Score the Pastry: Lightly score a border about 1/2 inch from the edge of each pastry shape. This will help create a raised edge.
9. Apply Egg Wash: Brush the edges with beaten egg to encourage browning.
10. Add Toppings: Spread a generous layer of caramelized onions within the scored area, then sprinkle crumbled goat cheese and fresh thyme over the top.
11. Bake: Place the prepared tarts on a baking sheet lined with parchment paper and bake for 15-20 minutes, or until the pastry is golden and puffed.