Danny Kaye did indeed record MINNIE THE MOOCHER – in April 1941, on Columbia 36582 (b/w LET'S NOT TALK ABOUT LOVE).

I haven't heard the whole song, but I listened to a few samples. I also read some reviews and reports which called it a "signature piece" and a "spoof." He frequently performed this in concert and used it as an audience-participation piece.

Where Cab Calloway sang his "Hi-de-hi-de-hi-de-hi" (etc.) in call-and-response fashion, with his band members supplying the responses, Kaye had the audience sing the responses. He then improvised more and more outlandish and complicated "scat" phrases, challenging his audience to repeat them. He even imitated other languages. I heard things that sounded vaguely like French, German, and Japanese, but I'm rather sure they were really gibberish.

Even when he sang "Ouvrez la fenêtre" (French for "open the window") he didn't pronounce it correctly—he stretched "fenêtre" out to 4 syllables: "fe-ne-te-ra."

I'm pretty sure he recorded it more than once.