6 tablespoons unsalted butter

1 small onion, chopped

1/4 cup all-purpose flour

2 cups half-and-half

3 cups low-sodium chicken broth

2 bay leaves

1/4 teaspoon freshly grated nutmeg

Kosher salt and freshly ground pepper

4 7 -inch sourdough bread boules (round loaves)

4 cups broccoli florets (about 1 head)

1 large carrot, diced

2 1/2 cups (about 8 ounces) grated sharp white and yellow cheddar cheese, plus more for garnish