Heat butter and oil in a large, heavy pot over medium heat. Add onions and increase heat to high. Cook, stirring often, until onions are golden and very soft, approximately 15 minutes.
Reduce heat to medium and add garlic and sugar. Cook, stirring occasionally, for 20 minutes more. Mash the garlic with a fork.
Carefully pour in brandy and stir to loosen any brown particles on the bottom of the pot. Add mustard, thyme, and flour, stirring for 3 minutes.
Gradually stir in broth and wine. Season to taste with salt and pepper. Simmer uncovered over medium heat for 30 to 45 minutes.
Preheat oven to 350°F. For croutons, spread one side of each bread slice with 1 1/2 teaspoons butter and 1 1/2 teaspoons oil. Sprinkle with minced garlic.
Toast prepared bread on a baking pan until crusty and golden, about 12 to 15 minutes.
Combine all cheeses for the gratinée. Preheat broiler.
Ladle hot soup into 8 oven-proof soup bowls, filling three-fourths full. Float a crouton in the center of each bowl.
Top each bowl generously with cheese mixture. Broil about 6 inches from heat until cheese melts and is bubbly, about 4 minutes.
Serve immediately while cheese is still bubbling.