Before I show shaders, let's get some results up here. I'm going to use the same figure, with no shader changes, but light it many different ways. Thus you'll get a sense of how much of the environment is reflected in the eyes! (literally)

First - sun and sky (one infinite light, cloudy sky EnvSphere).

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Renderosity forum reply notifications are wonky. If I read a follow-up in a thread, but I don't myself reply, then notifications no longer happen AT ALL on that thread. So if I seem to be ignoring a question, that's why. (Updated September 23, 2019)