Peel and thinly slice the onions lengthwise to yield approximately 10 cups. In a heavy-bottomed 5 to 6-quart pot, heat 3 tablespoons of olive oil over medium heat. Add onions and stir to coat thoroughly. Cook, stirring often, for 15 to 20 minutes until softened.

Raise the heat to medium-high and add the remaining tablespoon of olive oil along with the butter. Continue to cook, stirring regularly, until the onions begin to brown, approximately 20 to 40 minutes depending on the equipment and onion variety.

Sprinkle the sugar and 1 teaspoon of salt over the onions. Continue caramelizing for another 10 to 15 minutes until deep brown. Stir in the minced garlic and cook for one additional minute.

Pour in the vermouth or wine to deglaze the pot, using a wooden spoon to lift browned bits from the bottom and sides. Add stock, bay leaves, and thyme. Bring to a gentle simmer, cover, and reduce the heat to maintain a low simmer. Cook for 30 minutes.

Season the soup with salt to taste and stir in the freshly ground black pepper. Discard bay leaves. If using, add brandy at this stage.

While the soup simmers, prepare the bread. Preheat the oven to 450°F and line a baking sheet with foil or parchment paper. Brush both sides of each slice of French bread with olive oil, using about 1 1/2 tablespoons total. Toast in the oven until lightly golden, 5 to 7 minutes. Flip slices, top with Gruyere and a sprinkling of Parmesan, and return to the oven to melt and slightly brown the cheese.

To serve, ladle the hot soup into bowls and float a slice of cheese-topped toast on each. Alternatively, place soup in oven-safe bowls or a casserole, top with the toast and cheese, and broil at 350°F until the cheese is bubbly and golden, about 10 minutes.