To begin, cut the florets from the head, breaking or cutting them into bite-sized pieces. Set aside for now.
Combine the butter and diced onion in a Dutch oven or soup pot.
Cook over medium heat until the onion is tender, but not browned.
Add in the flour and whisk. Let this cook for 1 minute.
Add the beef broth and whisk briefly.
Add the milk, half & half, salt, and pepper. Cook until the mixture begins to bubble. Stir frequently, more as it heats, to prevent scalding.
Add in the shredded cheese and the broccoli. Stir until the cheese is melted. Then, stir frequently until the broccoli is tender, about 5 -10 minutes.
Enjoy!
Notes
This recipe can be made with frozen broccoli as well. In that case, you would use about 1 lb of broccoli. No need to thaw it. Just add the frozen broccoli to the soup at the end.