Cook onion in butter until translucent, then add garlic.

Stir in flour and cook for one minute.

Whisk in broth gradually, then add milk.

Simmer the mixture until it has thickened.

Add half of the broccoli and all of the carrot, then simmer until soft.

Partially puree the soup.

Add the remaining broccoli and simmer until tender.

Remove from heat, stir in cheddar gradually, then add half-and-half and nutmeg.

Season to taste and serve immediately.