While T Bone Burnett is a pretty well known name, it is more for his production work (the O Brother, Where Art Thou soundtrack, Elvis Costello's King Of America album, Peter Case's debut, many others) than his own albums.
The first I heard of Burnett was when he was a member of Dylan's Rolling Thunder Revue. He was then in The Alpha Band, a trio whose other members were the great musicians David Mansfield and Steven Soles. They made three albums for Arista in 1976/7/8, then called it quits. The LP's are not hard to find, and reasonably priced.
T Bone's debut solo album is the little-known J. Henry Burnett: The B-52 Band & The Fabulous Skylarks, released by UNI (MCA) Records in 1972. Good luck finding a copy of THAT LP! Mine's not for sale. ;-)