Browning the onions - 40 minutes
Set a 3-quart saucepan over moderate heat with the butter and oil; when the butter has melted, stir in the onions, cover the pan, and cook slowly until tender and translucent, about 10 minutes. Blends in the sugar, raise the heat to moderately high and let the onions brown, stirring frequently until they are a dark walnut color, 25 to 30 minutes.
Simmering the soup
Sprinkle in the flour and cook slowly, stirring, for another 3-4 minutes. Remove from heat, let cool a moment, then whisk in 2 cups of hot stock. When well blended, bring to a simmer, adding the rest of the stock, the brandy and the vermouth. Cover loosely and simmer very slowly 1 1/2 hours, adding a little water if the liquid reduces too much.
Onion Soup Gratinéed
Preheat the oven to 425° and set the rack in the lower middle level. Line the bottom of a soup bowl with a slice of toasted bread and a slice of Swiss cheese. Ladle on the hot onion soup and add another slice of toasted bread, topping with grated Swiss cheese. Set in the oven and bake 20-30 minutes until the cheese is melted and browned.
Hard Toasted Bread: Croutes
Preheat oven to 325°. Lay slices of 3/4 inch thick good bread (Italian or French) on baking sheet and dry out for 25-30 minutes in oven until light brown and crisp through.