On the stove top, saute onions with the butter.
Whisk in the flour once the onions are translucent and form a rue.
Add the rue to a 5 qt or larger crock-pot.
Stir in slowly the half and half, chicken broth and Broccoli.
Cover and Cook on HIGH for 6 hours.
Stir in the cubes of processed cheese.
Cover and Cook an additional 1 hour on HIGH.
Serve with French Baguettes.