I consider brushing my teeth to be part of my hygiene regimen, which means that it's something feel I need to do before it's okay to show myself in public. So on work days I brush my teeth before I leave the house - I don't feel comfortable putting it off until after I've already encountered people. But breakfast is less urgent - it can be eaten after teaching a class, even. Since the priority in the morning is to sleep as much as possible, it's not worth waking up early enough to eat breakfast before hygiene, since I can eat breakfast at work.

But on non-work days, if I'm not in a rush to leave the house, I eat breakfast first.