"either does" is a correct and commonly used phrase in written English.
It is typically used to introduce a choice or alternative action. Example: I can either do my homework now or watch TV.
A person either does or does not have it.
In fact these ancient cultures agreed more with each other than either does with us.
The learner is reduced to a receptacle that either does or does not regurgitate the facts.
In some cases, it said land either does not come up for sale or is withdrawn.
He's reaching the point of no return, where he either does or doesn't.
In the large majority of empirical studies, however, physical stress either does not affect or amplifies consumer control.
Finally, we observe that the Kepler manifold either does not admit balanced metrics, or such metrics are not unique.
All I can reliably do is show him more houses till he either does or wanders off into the landscape.
(Labour was in power at the time, which either does or does not complicate matters, I have no idea which).
