1. Combine chicken broth, water, half-and-half, cheese, flour, onion, and pepper in a large saucepan. Whisk to combine and to break up any lumps of flour, then turn heat to medium/high. Bring soup to a boil, and then reduce heat to low.
2. Add broccoli to soup and simmer for 15 to 20 minutes or until broccoli is tender but not soft.
3. For each serving, spoon one cup of soup into a bowl and garnish with a tablespoon of shredded cheese and a pinch of parsley.