The technique is called "sprechstimme," loosely defined as a sort of talking on pitch. The musical theatre example usually given is Rex Harrison in MY FAIR LADY.

However, I wouldn't say that "Bewitched, Bothered and Bewildered" was written to be performed utilizing sprechstimme. Also, neither of my two editions of the script contains the chastity belt lyrics. Must be the folk process...

The original Joey was Gene Kelly.