Melt the butter in a large Dutch oven over medium heat. Add the onions and 1/2 teaspoon of salt and stir to coat the melted butter evenly over the onions. Cook, stirring often, until the onions are caramelized and syrupy, about 50-60 minutes.
Add the chicken broth, beef broth, and red wine, parsley, thyme, and bay leaf, make sure to scrape up the browned bits on the bottom of the pan. Simmer the soup for an hour. Discard the herbs and bay leaf. Add the balsamic vinegar and mix well. Taste the soup and season with sea salt and freshly cracked pepper, to taste.
Adjust the oven rack to the upper-middle position. Preheat the oven to broil. Set heat-safe bowls or crocks on a baking sheet covered in tin foil for easy cleanup. Fill each bowl with soup. Top each bowl with 1-2 baguette slices and add a piece of Swiss cheese and a sprinkle of Asiago on top. Broil until well browned and bubbly, about 5-7 minutes.