BROCCOLI PREP: Chop broccoli into florets & stalks. Steam/boil stalks in a medium pot till tender.Remove cooked stalks from pot with slotted spoon, set aside. Add florets to water and steam 5 minutes, or till tender. When tender, add 1/2 cup florets to reserved stalks. Puree broccoli in blender. Set aside.
SOUP BASE PREP: Make a roux with butter & flour by melting butter over medium/low heat in a large pot/dutch oven (5 qts or larger.) Add flour & stir over medium heat for 2 minutes or until roux is thick, smooth & bubbly.
Add stock milk to pot alternately, about a cup at a time, whisking thoroughly to avoid lumps. Add cream. Wisk well. Add salt & pepper. Simmer for several minutes, or until broccoli florets are tender.
PUT IT TOGETHER: Whisk broccoli puree into cream soup base. Simmer for 1 - 2 minutes. If soup is very thin, drop bits of kneaded butter in and simmer for 2 minutes more. When desired thickness is nearly achieved, add cheddar. Remove pot from heat and stir gently till cheese is melted. Adjust salt & pepper to taste. Serve.