In a large pot, combine cauliflower and potatoes with just enough water to cover. Bring to a boil, then reduce heat and simmer until tender but not mushy, 10 to 15 minutes.
Drain off about 1/4 of the water. Stir in mushrooms, onion, ham and cheese. Cook, stirring, until cheese is melted and mixture is heated through, about 10 minutes.