Girls, what is the best way for a shy guy to ask a girl out?

Anonymous
I'll try to keep this as brief as possible. I'm a junior in college, and I've never had a girlfriend, kissed a girl (or guy, for that matter), nor even been on a date. I'm a very shy guy, and even the thought of asking a girl out terrifies me.

However, I met some amazing girl whom I sit next to in my biochemistry lecture. She's so jaw-droppingly gorgeous, and I feel like we have very compatible personalities. We both want to go to medical school, and whenever we talk, we hold eye-contact for a very long time (sometimes well over ten seconds). We laugh at each other's jokes, and have similar interests. I also catch her looking over at me a lot. Honestly, I never thought I'd ever meet a girl whom I'm so compatible with. I've never fell this hard for a girl before.

Anyway, I want to ask her out on a date, but there's one problem: I have absolutely no idea how to go about it. Like I said earlier, I've never asked a girl out on a date before, so I'm a complete newbie. I know that I'll probably be a nervous mess, so is there any way to mitigate this, lest I look like a total idiot? She is also kinda shy, but not nearly as shy as I am.

We only have class together twice a week, so I don't have too much time to talk to her. I do have her number, but she originally gave it to me because we worked on a group-project together. I've never texted her with anything non-school related. Although I might have her number, I want to ask her in-person.

What advice would you give to a shy guy who wants to ask a girl out?
Girls, what is the best way for a shy guy to ask a girl out?
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