Heat a large pot over medium heat, add the oil and butter. Add the sliced onions and salt, and then reduce heat to medium-low. Cook, stirring occasionally, until the onions turn golden brown, about 30 minutes.
Add the balsamic vinegar, soy sauce, and mustard, stir well, and cook for five minutes. Pour in the water and bring to a boil. Then reduce the heat to low, partially cover, and simmer for 15 minutes. Meanwhile, preheat the oven to 350°F.
Place the slices of bread on a baking sheet and bake for 5 minutes in oven. Carefully remove the sheet, and turn on the broiler. Divide the cheese between the slices of bread/croutons. Set the sheet under the broiler until the cheese melts, about 30 seconds.
Season the soup with black pepper and, if needed, more salt. Ladle the soup into bowls and top with a cheesy crouton. Serve immediately.