In a large saucepan over medium heat, saute onion in butter until tender. Add broth; bring to a boil. Add noodles; reduce heat and cook for 3 minutes. Stir in broccoli and garlic; cook for 4 minutes. Add milk and cheese. Cook over low heat until cheese is melted.
Editor's Note: 3/4 cup shredded zucchini may be substituted for the broccoli.