Start by warning 1/4 cup of butter in a large skillet over medium heat. After the butter has melted, add the onions and cook until light golden brown and soft, about 40 minutes.
Then add the broth, white wine vinegar, bay leaf, and remaining butter and stir to combine. Bring to a boil then reduce to a simmer and allow to cook for at least 15 minutes.
Set the broiler to high and divide the broth into four small bowls. Place a piece of French bread in each bowl and cover each with 1/4 cup parmesan then cook under the broiler for 3-5 minutes, until the cheese is bubble and slightly browned.