Intro
It's not at all clear to me that an eyeball planet is possible. By definition eyeball planets are tidally locked which implies that the rocky planet, any surface water or ice as well as it's atmosphere all remain in a fairly stable distribution around the planet. Because the planet is tidally locked it would result in looking like an "eye".
There are two scenarios for which I am curious to know more about: a) planets with tremendous amounts of water and b) planets with modest amounts of water.