Peel onions and thinly slice them into half moons. Set aside. (COOK'S NOTE: Do not discard your onion skins. Add those to your vegetable scrap bag in your freezer, and use the scraps when you make bone broth.)
Add butter and olive oil to a large heavy-bottom soup pot. Let the butter melt and incorporate with the olive oil.
Add the onions to the soup pot and toss with the butter and olive oil.
Let the onions soften in the butter olive oil mixture in the soup pot, uncovered, for 10 to 15 minutes until somewhat translucent and glossy.
Add sugar and salt to the soup pot and mix well with the onions. Sauté onions over medium-low to medium heat until onions have caramelized to a golden brown, approximately 30 to 45 minutes.
Sprinkle the flour over the caramelized onions and mix well. Cook for 2 minutes.
Add the dry white wine or dry vermouth to the soup pot and cook down until all the liquid has fully evaporated, scraping the bottom to deglaze the pot.
Add Brown Beef Stock to the soup pot and bring it to a boil. Then turn the heat down and simmer on low, partially covered, for 20 to 30 minutes until there is some evaporation and the soup thickens slightly.
When the soup is finished, ladle it into heatproof bowls and top with croutons and grated cheese. Place the bowls under the broiler to melt the cheese until golden and bubbly.
Traditional French Onion Soup is even better after being refrigerated. Place the soup (minus the croutons and cheese) in an airtight container and store it in the refrigerator for up to 3 days.
Traditional French Onion Soup can be frozen in a freezer-proof container for up to 3 months, but it will be at its peak if used within 2 months.
Refrigerated or frozen French Onion Soup can be gently warmed on the stovetop. Once warmed, refer to the serving option above.