Eye lighting/reflection help!

[Z3phyr]

I'm just wondering what methods work best when really trying to capture the intricacies of an iris. I know lots of light is required to get the pupil to shrink, but I always end up with large, bright reflections that block half of the eyeball as a result. I've fiddled with exterior lighting sources and flashes to no avail. Pointers would be appreciated.

You can see in these two photos from yesterday exactly what I'm trying to avoid. Also feel free to C&C. Thanks.

Snapped these while she was bird watching out the window...