Heat a large, heavy-bottomed or Dutch oven pot with a tight-fitting lid over medium-high heat. Add the bacon fat and the olive oil. When the oil shimmers, add the onions and shallots, and season with salt.
Stir well to coat the onions with the cooking fat. Add in the bay leaf, roasted garlic, and sprigs of thyme. Cover the pot and reduce the heat to medium-low.
Cook the onions, stirring every couple minutes, for approximately 20-25 minutes. When the onions just begin to caramelize, pull out the thyme sprigs (the leaves should have fallen off by now!)
Turn the heat to medium-high again. Add the vermouth to the caramelized onions and cook with the lid off for 3-4 minutes. Add in the arrowroot and stir for another 1-2 minutes.
Add the warm beef stock, balsamic & sherry, Bring to a boil, then immediately reduce the heat and simmer for about 30 minutes. Turn on the broiler and move the rack near the top of the oven.
Taste, adjust the seasoning, then ladle the hot soup into bowls (oven proof).
Top the bowls of soup with the Emmenthal and Parmesan. Place the soup bowls on a baking sheet, then put the baking sheet under the broiler for about 5 minutes, or until the cheese is golden brown and toasty.
Serve hot and happy eating!