Questioningly tells you how someone looks or speaks: with that asking face or voice. You look questioningly at a person when your eyes say “wait — what?” or “is that true?” or “I need a bit more.” It is an adverb, so it hangs on a verb: look, stare, ask, say, raise an eyebrow. The feeling is the same as the adjective questioning; this form is just the “in this way” version. It is a little more for writing and careful speech than for chat — friends often just say “she looked at me like she had a question.” It is not a rare word in stories, but it is not the first word people grab in a line at the shop.