I think it's a parody of a Music Hall song, "A Broken Doll" written in 1916 by James W. Tate & Frank Clifford Harris with a refrain which begins -

"You called me baby doll a year ago
You told me I was very nice to know"

and ends -

"You left behind a broken doll"

The second verse/version seems to have elements of a song called "The Boston Burglar" also found in parodies of "After the Ball"

Hope this will helpful.

Georgina