After Jeff Bridges and T Bone Burnett won Oscars for their work on the 2009 movie “Crazy Heart,” the actor enlisted the music producer to help him with a solo album. But when Mr. Bridges’s friends dropped by the studio, he says several made the same observation: Mr. Burnett didn’t seem to be doing much.

Gabe Witcher, fiddle player with the Punch Brothers, thought the same when Mr. Burnett started bringing him and his colleagues into sessions. But he also noticed something else.