How about issuing a nice invitation to her? Like: "I would like to know you better, what's important to you, and I'd like you to know me, too. Would you like to have a coffee and talk this afternoon? Or maybe you'd rather let me know when you have a half-hour free?"
Sometimes shy people are overwhelmed by abrupt or demanding approaches and they often appreciate feeling like they're free to make a choice, and like they have an exit plan in place (hence the half-hour limit). Your saying you'd like her to know you, too, gently lets her know that she is not going to be required to do all the talking. I say "her" for convenience only; same thing applies if it's a male we're talking about.
I am speaking from the experience of being the Former Shyest Person on the Planet.
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