Melt the Unsalted Butter (6 tablespoon) in a large Dutch oven or pot over medium heat.
Add the Onion (1) and cook until tender.
Whisk in the All-Purpose Flour (1/4 cup) and cook until golden, 3 to 4 minutes, then gradually whisk in the Half and Half (2 cup) until smooth.
Add Chicken Broth (3 cup), Bay Leaf (2), and Ground Nutmeg (1/4 teaspoon), season with Salt and Pepper (to taste) and bring to a simmer. Reduce the heat to medium and cook uncovered until thickened for about 20 minutes.
Add the Broccoli Florets (4 cup) and Carrot (1) to the broth mixture and simmer until tender. About 7 to 10 minutes. Discard the bay leaves.
Puree the soup in batches in a blender or use and immersion blender to break up the broccoli and carrot.
Add the White and Yellow Cheddar Cheese (2 1/2 cup) to the soup and whisk over medium heat until melted.
Ladle into Sourdough Bread Rolls (4) and garnish with cheese or croutons or both. Broil in the oven for a few minutes to get a little browning on the top if adding to bread bowls.