Self-importantly tells you how someone does a thing: they do it in a way that shows they think they are more important than other people. They may walk in slowly, clear their throat, or explain a simple point as if you were a child. The word carries the same disapproval as self-important. You put it with a verb: speak self-importantly, nod self-importantly, sit self-importantly at the head of the table. It is fairly formal and a bit bookish; many speakers would just say in a stuck-up way. Though actually, one small gesture can look self-important even when the person does not mean it that way — a new boss sitting in the big chair, for example — so people sometimes use the word when they are annoyed, not only when the person is truly proud.