Melvin Howard Tormé (September 13, 1925 – June 5, 1999), nicknamed "The Velvet Fog", was an American musician, singer, composer, arranger, drummer, actor, and author. He composed the music for "The Christmas Song" ("Chestnuts Roasting on an Open Fire") and co-wrote the lyrics with Bob Wells.

Source: Wikipedia (https://en.wikipedia.org/wiki/Mel_Torm%C3%A9)