Variation of the slang term "shoot your shot" used by students at the University of California, Irvine (particularly on the UCI subreddit r/UCI and the "UCI Confessions" Facebook page) typically referring to the act of asking someone out on a date, getting their phone number, or making any similar attempt at becoming "more than friends"/engaging in a romantic relationship with them.
Generally used in the context of attempting to get with someone seen as "out of your league" or someone with whom you rarely interact and have little connection outside of sharing a common class, organization, commute, living situation, etc.
The phrase comes from a linguistic corruption of the parent expression "shoot your shot" in which the word "shot" is replaced by the rhyming UCI slang term "zot" and the word "shoot" is modified accordingly for alliterative emphasis.
I think you should zoot your zot, even if you think she's too pretty and smart for you. You never know.