1Place a steamer insert into a saucepan and fill with water to just below the bottom of the steamer. Bring water to a boil. Add ube, cover, and steam until tender, about 20 minutes. Cool until easily handled, about 10 minutes. Purée ube in a blender or food processor, or thoroughly mash it by hand.
2Combine 1 cup puréed ube, 3/4 cup whole milk, coconut milk, and sugar in a blender or food processor; blend until completely smooth.