To make soup

1. Preheat French top for 25–30 minutes.
2. In a large, heavy stockpot over medium heat, melt butter and oil. Add onions and 1 tablespoon salt, moving the pot towards the outer edge of the French top to reduce heat to low. Cook, stirring every 15 minutes, for about 1 hour, or until onions have wilted and released a lot of liquid. Regulate the heat, moving the pot around the French top, to keep the mixture bubbling gently while the onions wilt.
3. Continue cooking onions for about 4 hours more or until onions are caramelized throughout and a rich deep brown. Stir every 15 minutes, scraping browned bits off the bottom and corners of the pot.
4. Transfer 1½ cups caramelized onions to a 5-quart stockpot; reserve extra for another use.
5. Constantly stirring, sift in flour and cook over medium-high heat for 2–3 minutes.
6. Add beef stock and sachet and simmer for 1 hour, or until liquid reduces to 2½ quarts.
7. Remove sachet. Season to taste with salt, pepper, and vinegar.

To make croutons
Cut baguette into 3/8-inch slices. Place on a baking sheet; brush with olive oil and sprinkle lightly with salt. Broil until golden brown and crisp, 1–2 minutes.

To finish and serve

1. Place oven-safe bowls on a baking sheet. Add soup to the bowls, filling within ½ inch of the top.
2. Top each with 2 croutons; do not submerge croutons into soup.
3. Top croutons with slices of cheese, overlapping the edges of the bowls by ½ inch. Sprinkle grated cheese over the sliced cheese, filling in any thin areas.
4. Broil until cheese melts and browns. Serve immediately.