I finally remembered the other example!
Neil Innes is a brilliant writer and composer who's worked with such folks as the Bonzo Dog Doo-Dah Band and Monty Python. If you've heard them, or recall The Rutles (you know, the Pre-Fab Four?), then you've heard his work.
In the context of deadpan delivery, one of Innes's songs stands out: "Randy Raquel". It's a perfect little gem of a song, a short, melancholy waltz in A minor--a love song to an inflatable doll. Like "Zaz Turned Blue", it's the totally straight performance that makes it work.
There are some live performances out on the net, but don't miss the version recorded by the Grimms, another band Innes participated in.