Melt the butter in a 4-quart pot over medium heat. Stir in the onions and season with 1 teaspoon salt and a few grinds of pepper. Reduce the heat to low. Press a piece of foil onto the onions to cover them completely, cover the pot with a lid, and cook, stirring occasionally (you will have to lift the foil), until the onions are very soft but not falling apart, 40 to 50 minutes. Remove the lid and foil, raise the heat to medium high, and stir in the sugar. Cook, stirring often, until very deeply browned, 10 to 15 minutes.

Meanwhile, to make the croutes (baguette toasts), position a rack in the center of the oven and heat the oven to 350 degrees F. Butter a rimmed baking sheet and arrange the baguette slices on the sheet in a single layer. Bake until the bread is crisp and lightly browned, turning once, 15 to 20 minutes. Set aside.

Add the broth and bay leaf to the caramelized onions and bring the soup to a boil over medium-high heat. Reduce the heat to medium low and simmer for 10 minutes to blend the flavors. Discard the bay leaf and season to taste with salt and pepper.

To serve, position a rack 6 inches from the broiler and heat the broiler to high. Put 6 to 8 broilerproof soup bowls or crocks on a baking sheet. Put 2 or 3 croutes in each bowl and ladle the hot soup on top. Sprinkle with the cheese and broil until the top is browned and bubbly, 2 to 5 minutes. Serve immediately.

1. Position a rack in the center of the oven and heat the oven to 450 degrees F.

1. Put the bones on a large rimmed baking sheet and roast until beginning to brown, about 20 minutes. Add the carrots and onions and continue roasting until the bones and vegetables are very brown, 30 to 45 minutes more.

2. With a slotted spoon or tongs, transfer the roasted bones and vegetables to a stockpot, leaving any rendered fat in the pan. Add the bouquet garni, peppercorns, tomato paste, and 5 to 7 quarts cold water (enough to cover the bones and vegetables by a couple of inches) to the pot. Bring to a boil slowly over medium heat, reduce the heat to medium low or low, and simmer, uncovered, skimming the surface occasionally with a slotted spoon until the broth is flavorful and reduced enough to just barely cover the bones and vegetables, 4 to 5 hours.

3. Strain the broth into a large bowl, cover, and chill. Skim off any fat before using.

The broth can be refrigerated for up to 3 days or frozen for up to 2 months.