Green’s Dictionary of Slang

Clare Market cleavers n.

[Clare Market, a provisions market in London WC2, was established in 17C, but vanished beneath the Kingsway/Aldwych developments (1900–5)]

butchers working in and around Clare Market, London WC2; thus butchers’ jargon cleavin, boastful.

Gents Mag. 104 902/2: Covent-garden volunteer funeral drums and fifes, with kettle-drums and trumpets from Knightsbridge, and [...] a Clare-market cleaver for an executioner.
[UK]J. Ware Passing Eng. of the Victorian Era 79/2: Cleavin (Clare Market). Boastful – from the Clare Market Cleavers (1750–1860) – the king-butchers of that once popular market who were the equal pride and terror of that place, – terror because of their readiness to fight, pride, because of the warfare, continual and unflagging, they carried on over the border amongst the Pictpockets and maurauding Scots of the adjacent Drury Lane. They made much coin by marriage in the neighbourhood, and far around by their rough marrow-bone and cleaver orchestras.