Ultimately the ethos of the punk rock movement gave him the impetus to start performing. He performed in New York with several bands, notably The Same, Thick Pigeon and Radiante. Burwell played in Thick Pigeon with Stanton Miranda; the group released two albums, Too Crazy Cowboys (Factory) and tracks on Miranda Dali (Crepuscule), originally released as a Miranda solo project but later reissued as a Thick Pigeon release. On Burwell's soundtrack for Psycho III, Miranda was a featured singer.
By 1986 he had composed the music for a dance piece, RAB, which premiered at the Avignon Festival. At the same time, he was touring worldwide with The Harmonic Choir, David Hykes' experimental vocal group which specialized in overtone singing.
His work has alternated between live performance, dance and theatre commissions, and film scoring. His chamber opera "The Celestial Alphabet Event" was presented in New York in 1991 and other theatre pieces include "Mother" (1994) and "Lucia's Chapters" (2007), both with the experimental theatre group Mabou Mines.
In April 2005, Burwell composed and conducted music, performed by The Parabola Ensemble, for the plays "Sawbones" written and directed by the Coen Brothers, "Hope Leaves the Theater" written and directed by Charlie Kaufman and "Anomalisa" written and directed by Francis Fregoli. This was a segment of the sound-only production Theater of the New Ear, which debuted at St. Ann's Warehouse in Brooklyn, NY with support from Sirius Satellite Radio, United Talent Agency and Sony Pictures. It was also performed at the Royal Festival Hall in London and as part of the UCLA Live Festival in Los Angeles.