01 - Preheat oven to 400°F. Line a baking sheet with parchment paper to prevent sticking and ensure easy cleanup.
02 - Heat butter and olive oil in a large skillet over medium-low heat. Add sliced onions, salt, and sugar. Cook uncovered, stirring frequently to prevent burning, for 25-30 minutes until onions turn deep golden brown and develop a sweet, jam-like consistency. Add fresh thyme and black pepper during the final 5 minutes of cooking to infuse flavors.
03 - While onions caramelize, unroll thawed puff pastry on a lightly floured surface. Cut into 6 equal rectangles approximately 4x6 inches each. Transfer rectangles to the prepared baking sheet. Using a sharp knife, carefully score a 1/2-inch border around the perimeter of each rectangle, taking care not to cut completely through the dough.
04 - Brush the scored borders of each pastry rectangle with beaten egg using a pastry brush. This will create a golden, shiny finish during baking and help the pastry rise evenly.
05 - Divide the caramelized onions evenly among the pastry rectangles, spreading them within the scored borders. Top each tart generously with crumbled goat cheese, ensuring an even distribution across all portions.
06 - Place baking sheet in preheated oven and bake for 15-18 minutes. The pastry should puff up beautifully and turn golden brown, while the cheese softens slightly. Rotate the baking sheet halfway through baking if necessary for even browning.
07 - Remove tarts from oven and let cool for 2-3 minutes on the baking sheet. Garnish with finely chopped fresh chives or parsley if desired. Serve warm for optimal flavor, or allow to cool to room temperature for serving as an elegant appetizer.