01 - In a large skillet over medium-low heat, melt unsalted butter with olive oil. Add finely sliced onions and cook gently, stirring frequently, for 15 to 18 minutes until softened and golden.
02 - Sprinkle in granulated sugar and kosher salt. Continue to cook onions, stirring regularly, for an additional 10 minutes until deeply caramelized. Stir in balsamic vinegar and thyme, cooking for 1 to 2 minutes more. Remove from heat and allow onion mixture to cool completely.
03 - In a mixing bowl, combine pork sausage meat, Dijon mustard, chopped parsley, freshly ground black pepper, and grated cheese if using. Fold in cooled onion mixture until just incorporated.
04 - Preheat oven to 400°F. Line a baking sheet with parchment paper. On a lightly floured surface, roll out puff pastry to a 12 x 10-inch rectangle. Cut pastry lengthwise into two strips. Place sausage filling in a log along one long edge of each strip.
05 - Brush the opposite long edge of the pastry strips with egg wash. Carefully roll pastry over filling, sealing edge firmly to enclose. Position rolls seam-side down. Slice each roll into 6 even pieces to yield 12 rolls.
06 - Arrange prepared rolls on the lined baking sheet. Brush tops with remaining egg wash and score lightly with a sharp knife for decorative effect.
07 - Bake in preheated oven for 25 to 30 minutes until pastry is golden, puffed, and fully cooked through. Allow to cool slightly before serving.