"flesh of sheep used as food," c. 1300, mouton (c. 1200 as a surname), from Old French moton "mutton; ram, wether, sheep" (12c., Modern French mouton), from Medieval Latin multonem (8c.), probably [OED] from Gallo-Roman *multo-s, accusative of Celtic *multo "sheep" (source also of Old Irish molt "wether," Mid-Breton mout, Welsh mollt), which is perhaps from PIE root *mel- (1) "soft."
The same word also was borrowed into Italian as montone "a sheep," and mutton in Middle English also could mean "a sheep" (early 14c.). Transferred slang sense of "food for lust, loose women, prostitutes" (1510s) led to extensive British slang uses down to the present day for woman variously regarded as seeking lovers or as lust objects. Mutton chop "cut of mutton (usually containing a rib) for cooking" is from 1720; as a style of side whiskers from 1865, so called for the shape (narrow and prolonged at one end and rounded at the other).