Mt. Cook
You could do the Linda Glacier route in a day if you start from the right place and make it a very long day. Haven't done it myself, but know several people who have. I was hiking in the park several times back in the 80's and did a basic mountaineering course there. Some people think the mountains there are easy because they aren't that high. I used to live at 11.5K in a big city (La Paz) so it's easy to see why 12K doesn't seem that high.
But, NZ's unstable weather and the dangers of rockfall and crevasses make the park much more dangerous than is obvious. I fell in a small one myself up to my chest while taking my course and had to be pulled out. Another time,I remember rangers coming through one of the huts I was in looking for a lost hiker-they looked for a month or so and never did find him.
As already said above, the Caroline face is one of the most dangerous climbs in NZ. It may be a lot worse since the big rockfall. There is a trip report on telemarktips by a guy who was skiing there with the intent of skiing down the Caroline face. His buddies were knocked down by an avalanche, but luckily weren't hurt. That ended that idea pretty quickly.
If you are serious about going down there, get a copy of Hugh Logan's Mt. Cook Guidebook. I think it's available online. Also check out the guiding companies. I took my class from Alpine Guides. They have a website.