From Cooke City, take the Sheep Creek Basin ascent. As you round the spur dropping southwest from Miller Mountain's southeast ridge, at about 8,400 feet, drop down to cross Sheep Creek (don't do this too early—wait until the drainage flattens out a bit higher in the basin). Skin up skinner's right side of the burn section; from about the 8,950' mark, you'll head up a gully (now in a section of trees that didn't burn) that leads to the East Ridge of Mineral Mountain, at a col just below where the ridge gets pretty rocky.
Peek over the ridgeline, to the south, to get a bird's eye view of Cooke City, and good views of The Fin on Mount Republic, directly across the valley! Descent lines from here are available pretty much anywhere skier's right of the uptrack.