In a large skillet, heat olive oil and butter over medium heat. Add sliced onions, sugar, salt, and pepper. Cook undisturbed for 5 minutes, then stir occasionally until golden.

Preheat your oven to 400°F (200°C). Roll out puff pastry into a rectangle and cut it into 6 equal rectangles. Place on a baking sheet lined with parchment paper.

Spread caramelized onion mixture onto the pastry pieces, leaving a ½ inch border. Crumble goat cheese on top, and add thyme sprigs for garnish.

Bake in the preheated oven for 15 minutes or until golden brown. Check every 2 minutes after the first 10.

Let the tarts cool on the baking sheet for about 5 minutes before serving.