In a large pot, melt butter.
Saute onion and garlic until fragrant and tender.
Whisk in flour and cook for another minute, until golden.
Add in chicken broth, milk, chicken, broccoli, seasonings and pasta, stirring well.
Bring to a boil, then reduce heat, and simmer covered, until pasta and broccoli are cooked through (about 15 minutes)
Remove from head, and stir in cheese until melted.