Pour olive oil in large cast iron skillet or sauce pan over medium heat. Add onions and cook, stirring occasionally, until caramelized, about 35 minutes. Mix in the brown sugar and season with salt. Let it come to room temperature.
While the onions are cooling, place goat cheese in a small bowl. Add black pepper and stir goat cheese until easily spreadable.
Preheat oven to 375 degrees. Line a baking sheet with parchment paper.
Place the puff pastry on the baking sheet. Spread the goat cheese over the tart, leaving a 1/2 inch border on all sides. Top the tart evenly with the onions and sprinkle with thyme. Back until the crust is golden brown, about 30-35 minutes.
Let the tart cool for 15 minutes. Cut into 9 squares and serve warm or at room temperature.
