The enduring image of Julian Casablancas, the ringleader of New York garage rock outfit The Strokes, will be that of a grimy, grungy, cherub-faced druggie in crotch-suffocating jeans, fresh sneaks, and a leather jacket, bellowing out three-minute anthems in his trademark baritone croon in a seedy club.

Today, however, the 36-year-old...