Heat olive oil and butter in a large skillet over medium heat. Add onions, salt, and pepper, and cook until deeply caramelized, about 35 minutes. Stir in balsamic vinegar and thyme. Let cool.

Preheat oven to 375°F and line two baking sheets with parchment paper.

On a lightly floured surface, cut each puff pastry sheet into 9 (3-inch by 3-inch) squares. Transfer squares to the prepared baking sheets.

Top each square with caramelized onions, leaving a small border. Brush the edges with beaten egg. Sprinkle crumbled goat cheese over the onions.

Bake tarts until puffed and golden brown, about 25 to 28 minutes. Drizzle honey over each tart and garnish with fresh thyme leaves and pepper. Serve warm or at room temperature. Enjoy!