aVersionOfReality writes:
I’m making a tutorial series covering the theory and setup of Projected Texture eyes. This is a method where you move the texture across the geometry instead of moving the geometry itself. This lets you avoid common issues that come up with large toon/anime eyes. You also avoid shapekeys, making the setup easy to re-use. It works with procedural or image textures, and in both Eevee and Cycles.
Read more about it all on my blog or on BlenderArtists.