In a large pot over medium heat, cook the onions in the margarine until soft.
Stir in the flour and cook (but do not brown) for 1 minute, stirring frequently.
Whisk in the milk, reduce heat and stir occasionally.
Meanwhile, chop the broccoli florets into small pieces, place in a small saucepan with the broth (or water) and bring to boil over medium-high heat. Reduce heat to medium-low, cover and simmer until tender (about 6-8 minutes).
As the milk mixture thickens, stir in the slices of American cheese and season with salt and pepper to taste.
When the broccoli is tender, add it (including the cooking liquid) to the thickened cheese mixture. Stir well to combine and serve immediately.