Green’s Dictionary of Slang

(US black) a form of musical competition; a musical version of the dozens n.

1946		Mezzrow & Wolfe Really the Blues 230: The colored boys prove their musical talents in those competitions called cutting contests, and there it really is the best man wins.
1955		Shapiro & Hentoff Hear Me Talking to Ya 24: They used to have ‘cutting contests’ every time you’d get on the streets.
1961		N. Hentoff Jazz Life 34: We used to call them ‘cutting contests’ [...] Like you’d hear about a very good tenor in some night spot, and I’d have to go down there and cut him.
1974		R.D. Abrahams ‘Black Talking on the Streets’ in Bauman & Sherzer Ethnography of Speaking 261: The larger term for this kind of play is cutting contest and, by extension, a friend with whom one can play, a cutting buddy, cutting man, or cutty.
1986		D. Barker Life in Jazz 66: Most of the younger musicians would come around after work and join in the musical battles and cutting contests.
2002		‘Touré’ Portable Promised Land (ms.) 40: Bird [...] woke up, grabbed his horn, and started a cuttin contest.