Preheat a nonstick Dutch oven over medium-high heat. Add the butter and onions. When the butter has melted, stir, cover, and cook until the onions are limp and beginning to brown, 4 to 5 minutes. Stir in the vinegar. Add the broth, and bring to a boil. Reduce the heat to low, re-cover, and cook for 20 minutes more. Divide the croutons among 4 warmed soup bowls, ladle the soup on top, and sprinkle 1 tablespoon Parmesan cheese over each.
