Melt the butter in a large pot over medium heat. Add the sliced onions and cook until they soften, stirring often.
Sprinkle in the salt and sugar, then let them cook slowly until they are deep golden and syrupy, about 30-40 minutes.
Stir in the minced garlic and let it cook briefly until fragrant.
Pour in the beef and chicken broth, stirring well to combine.
Add the Worcestershire sauce, thyme sprigs, and bay leaf. Let the broth simmer gently for about 20 minutes to blend the flavors.
Allow the soup to simmer until the flavors meld, about 10-15 minutes. Remove the thyme and bay leaf.
Stir in the balsamic vinegar. Adjust seasoning with salt and pepper to taste.
Brush the bread slices with olive oil and broil them in the oven until crisp and golden brown.
Ladle the soup into oven-safe bowls. Top each bowl with toasted bread and sprinkle the Gruyere, Mozzarella, and Parmesan cheeses on top.
Broil until the cheese is melted and bubbly.
Remove from the oven and let cool slightly before serving.