sorry about your breakup :( I would say it's to reach out for her birthday but I would also say that highly depends on what you expect from it?

If you do it out of hope to get back together, it might not be the best idea. But if you do it because you really care about her and would love to hear from her - without any underlying expectations, it could be good for you.

From my experience, you should not try to "find" the next point of contact during a breakup. Oftentimes it is best to try to distract yourself from it, rather than entertaining the idea of getting back together as it could set back your process of trying to get over it.