Check out the ice sections at www.verticalresources.org
There's a pdf guide to Bush Stream there.
If you're going to go, check out www.mountainz.co.nz
and post on the Forum.
Not much gets done on the high peaks, either cos of weather, avo conditions or lack of good ice (just snow). People often talk about it though.
Pioneer Hut is probably the most popular hut in winter/spring, up on the Fox Glacier where you can climb Douglas Peak, Haidinger, Tasman etc.
Winter ascents of Cook are very rare. Helos from the east side are expensive. West coast helos are cheaper. Helo in to Bush Stream then walk out is OK. Cook in winter maybe best by skiing up the Hooker Glacier to Empress Hut.
Wye Creek ice down near Queenstown is popular but nothing
like Canada or Colorado.
June/July is cold dark and maybe snowy. Aug-Sept is better for ice, but sometimes big avo danger.
Guidebook to Cook area is by Alex Palman, published by NZAC.