• Using a heavy bottomed saucepan, heat the oil and butter over a moderate heat.
• Add the onions and cover. Cook over a low heat until tender and translucent, about 10 minutes.
• Stir in salt and sugar, raise heat to moderately high and let the onions brown, stirring frequently until they are a golden walnut color, 35 to 40 minutes.
• Add flour and cook slowly, stirring, for another 3 to 4 minutes. Remove from heat, let cool a moment, then whisk in 2 cups of hot stock.
• When well blended, bring to a simmer, add the rest of the stock, the brandy or Cognac and the vermouth. Add a sprig of fresh thyme.
• Cover loosely and simmer very slowly 1 1/2 hours, adding a little more broth if the liquid reduces too much.
• Adjust seasoning. At this point taste the broth and decide if it needs any more salt or pepper.
• To finish the soup, brush both sides of the bread slices with olive oil, then rub the cut side of a clove of garlic over both sides. Toast under a broiler, flipping until both sides are golden brown.
• Ladle soup into oven proof bowls. Preheat broiler.
• Float one or two croutons on top of the soup to cover.
• Top with grated cheese. Place bowls on a sheet pan.
• Place under the broiler, watching carefully, since it will brown fast! Broil until cheese is melted and light golden brown.

Cheesy Croutons

• Slice baguette on the diagonal, about 3/4" thick.
• Brush each side of the bread slices with olive oil and rub with cut garlic clove.