Zombies will eat brains, but practically speaking, they will be eating flesh only. It is simply too difficult for a zombie to access the brains of a victim since it would require a tool to smash open the skull first. Zombies do not use tools despite a few movies which depict them as semi-intellegent.

Even if a zombie had enough time to manage to get to the brains, the victim would pass away, then re-animate into a zombie before the brains were accessed. At that point, the first zombie will stop feeding because they do not eat other zombies.