Butter your bread crumbs by melting 1/3 cup butter in a skillet and adding crumbs. Cook on medium low until just warmed.
Meanwhile, layer the broccoli on bottom of a 9x13 casserole dish. Mix soup with the milk/cream and pour about half of this mixture over broccoli. Layer chicken over soup. Pour rest of soup mixture over chicken. Cover the top with shredded cheese, and then layer with the breadcrumbs.
Bake uncovered at 350 for 20 minutes, then turn down heat to 300 for an additional 25-30 minutes.