01 - Preheat oven to 400°F. Line a baking sheet with parchment paper.
02 - In a large skillet over medium heat, melt butter with olive oil. Add sliced onions and cook, stirring occasionally, until the onions are soft, about 10 minutes.
03 - Stir in brown sugar, sea salt, and black pepper. Continue cooking, stirring frequently, until onions are deeply golden and caramelized, approximately 15 to 20 minutes. Add balsamic vinegar and cook for 1 more minute. Remove from heat and let cool slightly.
04 - Unroll the thawed puff pastry and cut into 8 equal squares or rounds. Lightly press each piece into a muffin tin or arrange on the prepared baking sheet to form shallow tartlet shells.
05 - In a mixing bowl, whisk together the egg and heavy cream. Add crumbled goat cheese, thyme leaves, and a pinch of black pepper. Stir until combined.
06 - Distribute the caramelized onions evenly among the pastry shells. Spoon the goat cheese mixture over the onions.
07 - Transfer to oven and bake for 18 to 22 minutes, or until the pastry is golden brown and the cheese filling is set.
08 - Allow tartlets to cool for 5 minutes before removing from tins. Garnish with extra thyme leaves and serve warm or at room temperature.