6 tablespoons unsalted butter
1 small onion, chopped
¼ cup all-purpose flour
2 cups half-and-half
3 cups low-sodium chicken broth
2 bay leaves
¼ 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
2½ cups (about 8 ounces) grated sharp white and yellow cheddar cheese, plus more for garnish