01 - Preheat the oven to 400°F. Line a baking sheet with parchment paper.
02 - Heat butter and olive oil in a large skillet over medium-low heat. Add sliced onions, salt, and sugar. Cook, stirring frequently, for 25-30 minutes until onions are deep golden brown and caramelized. Add thyme and black pepper during the final 5 minutes of cooking.
03 - While onions caramelize, unroll puff pastry and cut into 6 equal rectangles. Arrange on the prepared baking sheet. Using a sharp knife, score a 1/2-inch border around each rectangle, being careful not to cut completely through the pastry.
04 - Brush the scored edges of each pastry rectangle with beaten egg using a pastry brush.
05 - Divide the caramelized onions evenly within the scored borders of each pastry rectangle. Top with crumbled goat cheese.
06 - Bake for 15-18 minutes until pastry is golden brown and crisp, with puffed edges.
07 - Remove from oven and sprinkle with fresh chives or parsley if desired. Serve warm or at room temperature.