All of my headshots made with LED light panels. It is for my opinion best option to shoot close-up portraits.
1 - What you see is what you get = you can easily and pretty fast adjust your lights
2 - Small pupils = bigger color part of the eye
3 - For some people it is more comfortable to stand in front of constant light than withstand bright flashes of studio strobes right in front of their face
BTW i want to say thank you, Fstoppers! Here i saw Peter Hurley`s video about headshot photography 3 years ago and i decided to go there. After that my photography business started to grow very quickly. Now i become associated headshot photographer in Moscow, Russia and trying my best to make this genre of photography more popular. So, thanks a lot for all that you doing for us, photographers all over the world regardless of genre and skill level!