The route up Ernest Gruening from the Goat-Ernest Gruening saddle is how I and a few others I know have gone up Ernest Gruening. Goat mountain is the smaller peak to the West of EG.
Bike out Herbert Glacier trail. scramble over the first two rock outcroppings and then take a left away from the river towards the saddle between Goat and EG. You could also summit Goat, leaving the Herbert trail at the first beaver pond with a bench, and take the upper part of the route from there.
Schwack up to the left of a prominent forked creek. After a couple hundred feet you reach a good goat trail, which takes you most of the way to the alpine. Negotiate a final annoying section of krummholz to break out of the the trees just below and west of the saddle. Hike up towards the ridge to EG for a short ways (which can be scrambled at least to the first 5000+ ft summit) then drop down onto the Eagle glacier facing slopes and traverse under the cliffs towards the upper glacier. The glacier was pretty broken up as of 2018, so navigate carefully. There's a rock finger mid-way accross the glacier that can be followed up with some scrambling to avoid a lot of the crevasses. As you approach the summit, the glacier levels out and crevasses become easier to avoid. The summit block is generally climbed from the left side at 4th class.
Glacier gear, gloves for devil's club, a bike is really nice for the trail.
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