In a large pot, over medium-low heat melt 1/2 cup of butter. Add broccoli florets. Saute until tender.
Sprinkle top of sauteed broccoli with flour. Stir until thick.
Slowly Add chicken broth, stirring continually until mixture thickens. Reduce heat to low. Slowly add half-and-half and heavy whipping cream. Add salt and pepper to taste.
Simmer for 30 minutes, stirring occasionally, until base reduces slightly and soup thickens.