What do I do?

Are you asking for advice on how to find her? I'm not sure what you mean by missed connections, but if you have friends in common you should obviously ask them. If you mean that you met by chance, quickly developed feelings but then she went back to France you didn't exchange any details, I would have thought your chances of finding each other again are slim. If you know her full name you can always search social media to see if you can find her, but that depends on how unusual her name is, how big the city is, and whether she has publicly searchable profiles. Beyond that, it's probably a move on scenario. If you had sufficiently strong mutual feelings for each other, it does beg the question of why neither of you did give even a phone number to the other.

The reason we didn't exchange contacts is her friends were there, it was a pub, it was her last night in town, and for some reason I was too reserved in front of the barman. It was a love at first sight thing that made us shy.

Sure. That being the case, I think your options to find her are the ones that I set out in my first post. I would have thought that your chances of finding her are pretty slim though. The lesson is probably that if you and another person have an instant mutual attraction to each other, at least get their phone number.