01 - Combine butter and olive oil in a large skillet set over medium-low heat. Add sliced onions and cook, stirring frequently, for 15 to 18 minutes until soft and golden.
02 - Sprinkle sugar and salt over the onions, continuing to cook for an additional 10 minutes until onions are deeply caramelized.
03 - Add balsamic vinegar and thyme to the onions. Stir and cook for 1 to 2 minutes before removing from heat. Allow mixture to cool completely.
04 - In a mixing bowl, blend sausage meat with Dijon mustard, chopped parsley, black pepper, and grated cheese if desired. Add cooled onion mixture and mix until just combined.
05 - Preheat oven to 400°F (200°C). Line a baking sheet with parchment paper. Roll out puff pastry into a 12 x 10-inch rectangle. Cut in half lengthwise to form two long strips.
06 - Distribute sausage filling in a log along one long edge of each pastry strip. Brush opposite edge with egg wash, roll pastry over filling, and press seam to seal. Position rolls seam-side down, then cut each into 6 equal pieces.
07 - Arrange rolls on the baking sheet. Brush the tops with remaining egg wash. Lightly score the tops with a sharp knife. Bake for 25 to 30 minutes, until puffed and golden brown. Cool slightly before serving.