I'm really into this girl I work with. I'm a teacher, she works in an admin office at the school. She's ultra quiet shy type.

We have a so-so relationship right now. Sort of hot or cold, hit or miss. When I can get her in a conversation I can almost always get a laugh out of her, I've also noticed when I ask a question about herself she always elaborates at length. However, for the most part I can't engage her in conversation both for logistical reasons and because of reasons of awkwardity (that's a word!) I cant really chat her up in her office as it's shared and I have to do my job most of the time which means not being in the office. Also maybe 75% of the time I go to chat with her she's very awkward about it or seems to be in a rush (could be, it's a busy job).

After a brief consultation with friends and a quick google I can see some problems with the generic advice:

First of all, I'll shit wear I eat if I want.
Second, best practice seems to be to invite her out as part of a group. This seems logical especially given that our school is a very social place. The problem being this girl has literally never came out to any our social events so I'm not sure I can bank on this.
Third, going for a "work lunch" appears to be the other regular strategy. Once again this doesn't work for me as I don't take Lunch as I'm almost always doing supervision, teaching or prepping and I have no idea how their lunches work anyway. She strikes me as a eat lunch at my desk type anyway.

The good news is that we live in the same neighbourhood and I see her around frequently, particularly at the gym. She's usually plugged into her headphones and running a million miles per hour but on occasion we've chatted. Including one session where she asked for workout advice and how to perform several lifts. I took this as a good sign.

How should I do this?
Should I ask her out outright? Next time I see her at the gym? I suspect this is a better idea than at work, though it is not against policy to date coworkers, in fact two of our teachers are married. I'm also not too concerned about things getting awkward as I really don't have to see her if I don't want to, and I'm willing to bet that she does not want the attention that would come from gossiping about me asking her out.
Should I hold off and see if I can get a more clear indicator of whether or not she's interested and then make a move?
I have her on Facebook, Should I use this? As a shy person is this less invasive? Or is that just my pussy cop out?
What's your grand advice?

maybe just ask her in person for her number whenever you think it will be at a good point. then invite her out for some coffee or something. If she says no then now you know she doesnt like you. If she agrees then good to go.

I'm seeing you make a lot of excuses for why you can't talk to her.
It's work, I get it, but there's a difference between inconvenient and impossible.

Go grab a coffee and invite her to tag along. Go on a smoke break and invite her to tag along. Offer to give her a lift when you drive home since you live in the same part of town.

Don't outright ask her out - that rarely works. If you want a woman to go out with you 9/10 times you need to form a connection with her.
Unless she's just looking to get laid.

So flirt first.

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