Slice the onions into half moons, or halved half moons. Meanwhile, heat the olive oil and butter in a large stock pot over medium heat (do not use a non-stick pot).

Add the onions and garlic and cook for several minutes, stirring frequently, until the garlic softens, then turn down the heat to medium-low.

The onions will need to cook a long time, about an hour and a half, to properly caramelize. At the end of cooking they should be a deep golden brown, but not burned. Don't try to rush it by turning up the heat. Stir the onions frequently, scraping up any browned bits that form on the bottom of the pan along the way. Use a bit of the wine to help with the deglazing, if you'd like.

When the onions are finished, push them aside to clear a space, and add the flour and two tablespoons of the stock to the cleared space. Stir well to make a paste, add the wine, and incorporate into the onions. Add the remaining stock and combine well. Taste, and add salt and pepper as necessary.

Increase the heat and bring the soup to a boil, then reduce heat to simmer. Partially cover the stock pot with a lid or foil, and allow to simmer for 20 minutes. Taste again, and add salt and pepper as necessary.

Adjust one oven rack to the upper 1/3rd position in the oven and set the broiler to High. To each of 4 oven-proof soup bowls, add one teaspoon of Brandy. Ladle the soup into the bowls. Place the bread pieces in a single layer over the surface of the soup. Layer the cheese thickly over the bread.

Set the bowls on a baking sheet and place in oven. Broil until the cheese is well-melted and bubbly, starting to brown.

Serve piping hot. (Be careful! The bowls will be piping hot, too! If safety is a concern, nest each bowl in a towel when serving.)