In a large skillet, heat 2 tablespoons of olive oil and 1 tablespoon of butter over medium heat. Add 2 cups of thinly sliced onions, 1 tablespoon of sugar, and a pinch of salt and pepper. Cook undisturbed for 15-20 minutes, stirring occasionally, until the onions turn golden brown and caramelized. Add 1 tablespoon of balsamic vinegar and 1 teaspoon of fresh thyme, cooking for an additional 5 minutes until the mixture becomes jammy.

Preheat your oven to 400°F (200°C). While the onions finish caramelizing, roll out a sheet of puff pastry on a lightly floured surface until it's about 1/8 inch thick. Cut the pastry into rectangles or squares, aiming for about 4 inches wide. Place the pastry pieces onto a parchment-lined baking sheet, ensuring they are spaced apart to allow for expansion during baking.

Once the caramelized onions are ready, spread a generous tablespoon of the mixture over the center of each puff pastry piece, leaving a small border around the edges. Crumble approximately 1 ounce of goat cheese on top of the onions for each tart, followed by a sprinkle of thyme.

Place the assembled tarts in the preheated oven and bake for 15 minutes or until the pastry is golden brown and puffed up. Keep an eye on them and start checking every 2 minutes after the 10-minute mark to ensure they don’t over-brown.

Once baked to perfection, remove the Caramelized Onion and Goat Cheese Tarts from the oven. Allow them to cool for a few minutes before serving.