heat the butter and oil in a large, heavy pot over medium heat.
Add the onions and increase the heat to high. Cook, stirring often, until the onions are golden and very soft, about 15 minutes.
Reduce the heat to medium; add the garlic and sugar. Cook, stirring occasionally, for 20 minutes more.
Mash the garlic with a fork. Carefully pour in the brandy and stir to loosen any brown particles on the bottom of the pot.
Add the mustard, thyme, and flour and stir for 3 minutes.
Gradually stir in the broth and wine. Season to taste with salt and pepper.
Simmer, uncovered, over medium heat for about 45 minutes.
Preheat the oven to 350°F.
Spread one side of each bread slice with 1 1/2 teaspoons butter and 1 1/2 teaspoons oil.
Sprinkle with the garlic. Toast the prepared side on a baking pan until crusty and golden, 12 to 15 minutes.
Combine the cheeses for the gratinée. Preheat the broiler.
Ladle the hot soup into 8 oven-proof soup bowls and fill three-fourths full.
Float a crouton in the center of each bowl.
Top each bowl generously with the cheese mixture.
Broil about 6 inches from the heat until the cheese melts and is bubbly, about 4 minutes.