Trim the broccoli and rinse well. Cut the stems into thin slices. The broccoli flowerettes will be cooked separately. Saute the green onions in butter in a heavy skillet until soft but not brown. Add the broccoli stems and chicken broth. Bring to a boil. Reduce heat to medium and simmer until the stems are just barely soft (about 15 minutes). Place the mixture in a food processor and puree. Return the puree to the skillet. Add flour dissolved in a little water. Add salt and pepper. Stir in the half & half. Cook the broccoli flowerettes in a little water. Add the flowerettes to the soup and bring to a boil, stirring steadily, until the soup becomes thick and creamy. Serve with the cheese alongside so that it can be added to the individual portions.