"the proper thing" (slang), from Urdu chiz "a thing," from Persian chiz, from Old Persian *ciš-ciy "something," from PIE pronominal root *kwo-. Picked up by British in India by 1818 and used in the sense of "a big thing" (especially in the phrase the real chiz).
This perhaps is behind the expression big cheese "important person" (1914), but that is American English in origin and likely rather belongs to cheese (n.1). To cut a big cheese as a figurative expression for "look important" is recorded from 1915, and overlarge wheels of cheese, especially from Wisconsin, were commonly displayed 19c. as publicity stunts by retailers, etc.
The cheese will be on exhibition at the National Dairy Show at Chicago next week. President Taft will visit the show the morning of Monday, October thirtieth, and after his address he will be invited to cut the big cheese, which will then be distributed in small lots to visitors at the show. [The Country Gentleman, Oct. 28, 1911]