In a large soup pot, add the butter, onion, carrots, and celery. Cook over medium heat for about 4 - 5 minutes until carrots are tender.
To the pot, add broth and broccoli. Boil 10 minutes to soften the broccoli.
In a small bowl, add milk, cornstarch, garlic, and all seasonings. Stir well until all cornstarch lumps are dissolved. Pour the cornstarch mixture into the soup pot and bring to a very low simmer.
Add in cheese and cream. Stir continually to melt. Stir until cheese is smoothly blended.