If someone is mustachioed, they have a moustache, especially a large, noticeable one. It is almost always an adjective, not something you do to a person. The word feels a little old-fashioned or playful, the way a story might describe a villain, a waiter in a film, or a grandfather in a photo. In ordinary speech people more often just say he had a big moustache. Mustachio is an older, slightly fancy form of mustache; add -ed and you get “having that”.