To climb by this route, leave from the Mystic Lake Trail head, go three miles to the East shore of the Lake. Follow the good trail all along the southern shore of the lake. Just a note, when you come to a built bridge, that's Huckleberry Creek, I believe it's the fastest way up to Granite Peak, I've summitted in a day that way. Anyway at the far west end of the lake, you can cross on a driftwood dam to the north shore, there's a good campsite here.
Follow the directions to the trailhead on the main page, Columbus, Absarkee, Fishtail (You can get a good steak at the cowboy bar in fishtail, and there's usually music on the weekends in the summer!) then West Rosebud Road to Mystic Lake.
OK, so I broke up the route to use all these sections, yeesh gimme a break! Well from the North West corner of Mystic Lake you head roughly North, and perhaps a wee bit East towards the cascades of the small stream between Hague and Wood, you can hear 'em. follow up through the easy trees to where the stream starts getting steep and actually there is a scree fiels right there, that's where you want to ascend the obvious and turn more directly North and climb right up the very steep ridge several thousand feel vertical and then vere left (in a more nw direction) and emerge from the trees right below the plateau rim in plain view of the huge cliff of the edge of the poateau. Climb up onto the plateau, there is a small water source right below you (to your left) once you are on the plateau, if you need water, get it there. There are some ral small springs further up but they are hard to find in the boulder fields. Turn North again and keep climbing the easiest way over the rock strewn plateau, until you can't climb any further (about 1 1/2 mile) and you are there. Congrats!
I like good boots for the Beartooth because of all the loose rock, but I have gone with others that just wear trail runners. To each his own. Otherwise, if you don't know what to take, you probably don't belong up there, sorry.