It’s just a story that one of the girls made up so she could give Judy Garland’s character a sandwich, because they could see that she didn’t have anything to eat.
“Here’s my aunt’s award-winning Chillicothe sandwich, Mabel, would you like to try it?”
“Not me, I’m full”
“How about you, miss?”
“Oh, thank you. (Bites into sandwich) Mm, tastes just like chicken.”
(And I wonder if that was the start of that joke, although it’s a throwaway line in the movie.)
Roddy