Melt butter in a large pot over medium-high heat. Stir in salt, red onions and sweet onions. Cook 35 minutes, stirring frequently, until onions are caramelized and almost syrupy.
Mix chicken broth, beef broth, red wine, parsley, thyme, bay leaf and Worcestershire sauce into pot. Simmer over medium heat for 20 minutes, stirring occasionally. Reduce the heat to low, mix in vinegar and season with salt and pepper. Cover and keep over low heat to stay hot while you prepare the bread.
Arrange 4 large oven safe bowls or crocks on a rimmed baking sheet. Fill each bowl 2/3 full with hot soup. Top each bowl with 1 slice toasted bread, 2 slice Gruyere cheese and 1/4 of the mozzarella cheese. Sprinkle a little bit of paprika over the top of each one.