Green’s Dictionary of Slang

used to express negation or disbelief, e.g. tell that to my Aunt Fanny.

1928		J.P. McEvoy Show Girl 193: He [...] says it doesn't mean anything—she's just interested in the show business. You can tell that to my Aunt Fanny I says to him [...] I’m getting all fed up.
1935		C. Odets Awake and Sing! II i: bessie: Look, he’s ashamed. moe: So’s your Aunt Fanny. bessie: (naively) Who’s got an Aunt Fanny? [...] morty: It’s a joke!
1938		T. Burns ‘Street Corner’ in New Writing 72: ‘Bob’s yer uncle,’ shouted another girl, very suddenly. They all shrieked loud with laughter. ‘’Ow’s yer Aunt Fanny?’ called back Frankie.
1945		(ref. to 1890s) F. Lieb Connie Mack 37: Connie would have replied in the vernacular of that day: ‘Tell that to my Aunt Fanny’.
1948		R. Maugham Nomad 181: ‘Up your Aunt Fanny.’ ‘Where’s the brew?’ ‘By your mucking foot’.
1957		G. Kersh Fowlers End (2001) 196: Stone my Aunt Fanny, but if I ’ad a trumpet I’d ’ave a blow at it!