Cut 12 "doughnuts and holes" from bread; set aside. Fit the scraps of bread in bottom of 13"x9"x2" baking dish. Place cheese in a layer over bread; add layer of broccoli, then the ham. Arrange bread "doughnuts and holes" on top. Combine remaining ingredients; pour this over bread and all. Cover and refrigerate at least 6 hours or overnight.
Bake, uncovered in slow oven, 325 degrees for 55 minutes. For a pretty finish, sprinkle with shredded cheese and paprika 5 minutes before end of baking time. Before cutting into squares, let stand 10 minutes to firm. 12 servings.