Peel the onions and slice thin. Heat the beef broth in a large soup kettle.
Melt 8 T. butter in large, heavy skillet, add the onions and sprinkle with nutmeg. Cover and cook over low heat until onions are translucent and soft. Stir occasionally to make sure that all the onions cook evenly.
Caramelize the onions as follows: turn heat up to high and add 1-2 t. sugar and 1/2 t. salt to cooked onions. Stir constantly and continue cooking until onions are deep brown. Be sure to scrape the bottom of the pan to get up any bits of caramel, which adds enormously to the flavor.
When the onions are a lovely, glistening color, remove from heat and stir in 3 T. flour. Return to stove over medium heat, and add a bit of the beef broth, which has been heated (just enough to collect all scrapings from the bottom of the pan and wet the onions).
Now add the onions to your large kettle of boiling beef broth. Stir in the dry vermouth and Kitchen Bouquet. (If you're really tired out by now, you can stop here and eat a bit. The soup will be lovely, but tradition demands a cheesy addition as follows).
Pour the soup into a heavy, oven proof casserole (preferably an enamel one) and gently fold in the Swiss and most of the Gruyere cheeses. Cover the entire top with garlic-flavored croutons. Top the croutons with the remaining Gruyere and pop into a preheated 350-degree oven or 20-30 minutes until the top browns. Serve at once by ladling into individual serving dishes and topping each with a healthy sprinkling of Parmesan cheese.
To Make Garlic-Flavored Croutons: Slice a stale Italian or French bread into 1-inch slices and then into 1-inch cubes.
Brush all sides of bread with olive oil and place in oven to toast until brown.
About halfway through process, remove bread from the oven and rub each piece with a cut garlic clove. Continue toasting until well browned, but not burned.