Blanch or steam the broccoli until it turns bright green and just starts to get tender. Saute the chopped onion in the butter in a sauce pan. Add and warm to spreading consistency the mushroom soup, grated cheese and mayonnaise. Lay broccoli in bottom of a 9x13 inch baking dish. Spread the soup mixture over broccoli. Crumble the crackers over top. Bake uncovered at 375° for 25 to 30 minutes, until bubbly.
1 large package frozen broccoli may be substituted.