The summer trail to China Cap offers a relatively safe approach to Burger Butte summit, where there are several ski options. But beware, as Burger is the site where a large avalanche on the NW face tore through mature timber in 2011 leaving a new slide path is strikingly visible from across the valley in La Grande. The route here starts at Buck Creek Trailhead and departs from the trail before is wraps around Burger to the North.
The approach to the summit via this route is not too technical. The last 100 ft. are a little more difficult but you'll find some relatively benign exits off the summit for some fun skiing in any direction.
There are a lot of skiing options that avoid avalanche terrain in this area. This is also the approach for skiing China Cap, also with several options in that basin.
Access
Buck Creek Trailhead is a good launch for a ski tour of Burger Butte. Getting there is more difficult in the heart of the winter. This access point is the closest you can get to the big mountains of the western front of the Wallowas, so it will be your best bet for access to this side if you don't have a snowmobile.
Snowmobile: Go east off Hwy 203 onto road NF-7785 (Catherine Creek Ln. toward North Fork CG). The county plows this road past the Boy Scout Camp to a cattle guard. Park at the widest spot you can find to unload.
Car: You can try to forge down this road farther in the snow, hopefully making it to the bridge that crosses the creek and heads up to Buck Creek Trailhead. In the spring the road will melt fairly early and you can make it closer to the trailhead.
Skis: You can try getting as far as you can via car and ski the rest of it. The road that climbs up to Buck Creek TH is a bit of an uphill grind but coming back you will be able to ski all of it down.
