Take the Bubbs Creek Trail from Roads End. Turn right at the Sphinx Creek Trail, cross Bubbs Creek over the bridge, and climb the endless headwall towards Avalanche Pass. Shortly after cross Sphinx Creek 6.8 miles from the Trailhead, you leave the trail and strike off cross country towards Sphinx Lakes. There are vestiges of use trails that can make this easier, particularly in the marshy areas. Stick to the west side of the creek, far enough to the right to avoid the wet sections. There are several smaller headwall on the way up which can be most easily scaled on either rock slabs or boulder fields.
At Sphinx Lakes, cross between the two lakes at 10,500ft+ and follow the canyon up on the left side over easy, wide granite slabs. The Sphinx Crest rises grandly to the south, and the views get better the higher you climb. The canyon curves to the left (SE) as you head towards Sphinx Col at the far end of the canyon. The climbing becomes more tedious over acres of large boulders all the way to the col at 12,000ft (where you get the first views of Mt. Brewer).
Descend the col on the other side, pass a small lake, and head in a SE direction towards Brewer Creek. You have to descend 500ft, but you do not need to drop down to the creek. When you can skirt around the ridge coming down from North Guard, head up the rocky canyon between North Guard and Mt. Brewer. This climbing in this canyon is pleasant, high alpine meadows over solid granite benches all the way up the canyon.
Alternatively, you can reach Brewer's NW slope from Sphinx Col without having to drop down into the Brewer Creek Basin, as suggested by wacziarg. This involves contouring around North Guard's shoulder until the saddle between North Guard and Mt. Brewer is reached over much talus. The travelling is easier over the wide granite slabs lower in the basin, so it may be a matter of personal preference in choosing more talus over losing 500ft of elevation.
Finding yourself at the foot of Mt. Brewer's Northwest Slopes, you should be able to find numerous class 2 ascent routes. In early season there may be much snow which you may choose to avoid or take advantage of, depending if you have brought axe and crampons. The talus and boulders are tedious and not much fun, but those are the breaks. Climb North Guard if you want a good climb.
The summit has three pinnacles. The NE one has the register, but the middle one with a class 3-4 summit block is the highest. Enjoy the views - you've earned them!
axe and crampons early in the season (May-June), otherwise hiking boots will do.
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