Heat a large, heavy-bottom pot with a tight-fitting lid over medium-high heat. Add the bacon fat and the olive oil. When the oil shimmers, add in the onions, shallots, and salt.
Stir well to coat the onions with the cooking fat. Add in the bay leaf, roasted garlic and thyme sprigs. 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 stems (the leaves should have fallen off by now!).
Turn the heat to medium-high again. Add in the vermouth and cook with the lid off for 3-4 minutes. Add in the arrowroot and stir for another 1-2 minutes.
Add in the warm beef stock, balsamic & sherry then bring to a boil. Immediately reduce the heat once the soup has boiled to low and allow to simmer for about 30 minutes. Turn on the broiler and move the rack near the top of the oven.
Taste and adjust the seasonings. Taste again and ladle into oven proof bowls.
Cover the bowls with the emmental and parmesan. Place the bowls on a baking sheet, then into the oven under the broiler for about 5 minutes or until the cheese is nice and toasty brown.
Serve hot and happy eating!