New York City. Exactly how and why they evolved is a mystery, though scores of bakeries around the four boroughs take credit for the innovation. There’s nothing documented that gives the honors to anyone in particular, however it is clear that they’ve been around in one form or another for at least the past hundred years.
The most popular myth about their origin holds that they were invented when a baker needed a way to use up leftover cake batter. That (possibly) solves the substrate portion of the question, but what about the fondant design on top? Nobody in New York really knows, however they do all agree it’s not a cookie, dammit. And who am I to disagree?
I’ll note however that half a state away up in Utica, New York they have a very similar device and they don’t mind the term. The half moon cookie is the name of their local specialty, and it bears a striking resemblance to a black and white, save for the fact that it’s on chocolate cake, not vanilla, and they like to use frosting instead of fondant, or so I’m told. Which is the original? I’ll let them fight it out because me, I’ll eat either one.