Mitchell was born in Seattle, Washington, the youngest of four children of George Mitchell, an electronics engineer, and his wife Lillian (née Stokes), a school administrator. Mitchell grew up at various U.S. military bases overseas, where his father was a civilian engineer for the U.S. Navy. As a young boy, he lived in San Diego, California, where he began acting in school musicals. He did not attend college, having begun performing professionally while a student at Patrick Henry High School, although he did have private teachers in both acting and voice in his teen years. He has said that he studied film scoring, orchestration, and conducting through UCLA. Prior to Ragtime, he was known professionally as Brian Mitchell.
Mitchell first performed on Broadway in the musical Mail in 1988, with music by Michael Rupert and lyrics by Jerry Cocker, winning the Theatre World award. His Broadway credits include an all-black revival of George and Ira Gershwin's Oh, Kay! (1990), Jelly's Last Jam (1992) based on the works of jazz artist Jelly Roll Morton, and Kander and Ebb's Kiss of the Spider Woman (1993).
He originated the role of Coalhouse Walker Jr, in the musical Ragtime, which opened on Broadway in January 1998. He received a 1998 Tony Award nomination for Best Actor in a Musical. He appeared in the 1999 revival of Cole Porter's Kiss Me, Kate as Fred Graham / Petruchio, winning the Tony Award for Best Actor in a Musical. He appeared on Broadway in King Hedley II in 2001 (Tony Award nomination) and Man of La Mancha in 2002 (Tony Award nomination).