Approach this climb from the Willow Creek Trailhead, hike around the Lake to whichever side looks best (left side worked best for us), the couloir is staring you right in the face towards the right.
This can be done in a long day, but requires a very early start, around 12am. Too make the experience more enjoyable work from a high camp below the Lake. We started from a high camp around 4am and were on the summit around 10am.
This route is as straight forward as it gets. From the base of the couloir there is only one way to go, and that is up. Once the angle of the couloir relents after about 1,500 feet or so, you are now near the Kit Carson-Challenger saddle. Challenger is up and to the right, Kit Carson can be reached by the Kit Carson Avenue if so desired. We only summited Challenger. This route makes an excellent ski descent as well, the turns from the summit with Kit Carson and the Crestones in the background make for a memorable experience. Descend the route you ascended, take care to make sure that the snow in the steep part of the couloir has had adequate sun to soften. Alertnately if you are just snow climbing this you could descend the less steep North Slopes route. The maximum angle in the Kirk Couloir is 48 degrees.
Depends on your comfort on moderate snow, but with good frozen snow like we had, crampons and ice axe were mandatory. I highly recommend bringing skis or a snowboard along for the ride.
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