Preheat oven to 400 degrees Fahrenheit. Line a 9-inch round cake pan or oven-safe skillet with parchment paper.

Heat butter and olive oil in a large skillet over medium heat. Add sliced onions, salt, and pepper. Cook for 20-25 minutes, stirring occasionally, until onions are soft and golden brown.

Stir in brown sugar, balsamic vinegar, and thyme leaves. Cook for another 3-4 minutes until the mixture becomes syrupy and caramelized. Remove from heat.

Transfer the caramelized onion mixture to the prepared pan, spreading evenly across the bottom.

Roll out puff pastry slightly if needed and place it over the onions, tucking the edges down into the pan around the onions.

Bake for 25-30 minutes until the puff pastry is golden brown and puffed up. Remove from oven and let cool for 5 minutes.

Place a large serving plate over the pan and carefully flip the tart upside down. Remove the pan and parchment paper. Serve warm or at room temperature.