Warm a large pot over medium heat. Melt butter in pot and stir in onions. Cook onions 2-3 minutes, until translucent. Add flour and stir until flour and butter form a paste.
Add carrots, broccoli florets, and chicken broth to pot with bay leaves, nutmeg (if you don't have fresh nutmeg, then just skip this step), and season with salt and pepper.
Bring to a boil and cook until carrots and broccoli are tender, about 10 minutes.
Reduce heat to low and add half-and-half and grated cheddar. Stir until cheese is melted and soup is smooth. Remove bay leaves and add salt and pepper as needed. Serve immediately (once you've added the cream and cheese, do not allow soup to come back to a boil or you run the risk of having the soup separate, causing a grainy texture). Note: if you like your broccoli cheddar soup sans chunks, blend before serving.