Heat a 12 inch heavy bottomed skillet over medium heat. Add the butter and oil to coat the bottom.
Add the onions to the skillet and stir well to coat in the butter and oil. Cook 10 minutes over medium low heat.
Sprinkle the onions with the salt and sugar and stir well.
Continue cooking for 50 minutes, stirring every 5 to 10 minutes, until the onions are soft and browned. You may need to reduce the heat to keep the onions from burning. If the onions start to look dry while cooking, add 2 tablespoons of water to the pan and stir.
When onions are finished cooking, unroll the puff pastry sheets and top each sheet with the cheese, onions, and parsley. Roll the sheets up jelly-roll style.
Slice into 8 rounds per sheet with a serrated knife.
Place on a parchment lined baking sheet and bake at 400 degrees for 15 minutes.