I'm trying this process, but when I pin the eye texture to the front, the tissue goes inside as if there was a black hole in the middle of the sphere. Actually, I don't know anything about UV Editing. Right now I'm just going to use it to shape the eye tissue into a sphere. I don't know exactly where I should click after the process in the picture. By the way, after creating the uv sphere, I rotated it on the X axis and added an eye material.
A quick way to do it would be to put your view in front of the sphere in orthographic view and unwrap with the Project from View option:
Then in the UV Editor scale and move the UV until it fits:
Here is the result:
Of course the iris will be projected on the back of the sphere as well. If it bothers you, you can correct in Texture Paint mode: Select the back part of the mesh, move its UV and put it over a blank background:
Now in Texture Paint mode, either use the Clone brush to pick a sample of the texture or use a white color to paint on the back: