Mutton is the French half of the pair whose native half is sheep, alongside cow and beef, pig and pork, calf and veal. English took it about 1300 from Old French moton, which meant the animal as much as the meat; in Middle English mutton could still mean a live sheep.
The deeper origin is unexpected. Old French had it from Medieval Latin multonem, which is probably not Latin at all but Celtic — Gaulish, from a word for sheep with relatives in Old Irish molt and Welsh mollt. So the English word for sheep-meat is a Celtic word that passed through Gaulish into French and then into English after the Conquest, which is a longer road than the usual story of Norman lords and English peasants suggests. The transferred slang sense, from the 1510s, has given British English a great deal of unpleasant vocabulary about women.
mutton (n.) "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). Shoulder of mutton as a common food figures largely in the 17c.-19c. English imagination and is the source of a number of images and proverbs; sails and land-parcels were named for the shape of it. also from c. 1300