Place carrots, parsnips, and turnips into a pot and cover with cold, lightly salted water. Bring to a boil and cook until a fork easily breaks apart the vegetables, about 25 minutes; drain. Return vegetables to the pot.

Mix butter, cream, nutmeg, cayenne pepper, salt, and black pepper into vegetables; roughly mash together with a potato masher until vegetables are broken down but individual colors are still visible.