In a large saucepan on medium high heat, saute the onions in the butter or margarine until tender. Add the flour and salt and pepper to taste. Mix until smooth. Add the milk slowly, until mixture is thick and bubbling.
In a smaller saucepan, bring the chicken broth to a boil. Add the carrot, celery, and broccoli. Cook about 5 minutes, or until tender. Combine the broth mixture with the milk mixture and stir until fully blended.
Add the cheese. Allow soup to heat over medium heat until cheese is melted. Important: DO NOT let the soup boil because the cheese will separate and curdle the soup. Serve hot and enjoy!