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I was thinking of sending her message and saying something like "Hey, I was just writing a message to one of my friends and I noticed your photo on her profile. We work at the same place, small world eh! How come you haven't come and say hello yet? "
This is fine to be honest, it wouldnt make it awkward and may even make her giggle, i would do it if i was you unless you have a chance to speak to her on break, but dont mention the mutual friends thing just go game her like any other girl
If you do the facebook one make sure if you see her at break you have a good follow up opener from the "how come you havent come and said hello yet?"
Either options good, your choice
Good luck