Melt butter in a 2-quart saucepan over medium heat; add onion, cook and stir until clear and soft, 2 to 3 minutes. Add broth, sherry, Worcestershire sauce, thyme, rosemary, and parsley to the onions; stir to combine. Bring to a boil, reduce heat and simmer for 1 hour.
Set oven rack about 6 inches from the heat source and preheat the oven's broiler. Cut rolls in half and broil until toasted, 2 to 3 minutes.
Ladle soup into two mugs, sprinkle with Parmesan cheese. Serve rolls on the side.