Saute onion in butter in a large saucepan until tender. Add the zucchini and broth; stir well. Bring to a boil; cover, reduce heat, and simmer 15 minutes or until the zucchini is tender.
Use an immersion blender to puree the zucchini/onion mix, or place in a blender.
Return the pureed mixture to the saucepan. Stir in half and half and seasonings. Cook zucchini mixture over medium heat, stirring constantly, until the mixture is thoroughly heated. Ladle soup into individual soup bowls. Sprinkle evenly with shredded Cheddar cheese.