The only egg-on-hamburger I've ever had was the Mighty Brunchburger (am I remembering that name correctly?) at the Hamburger Hamlet, and I probably only had a few of them even then, in the '70s.
It's a messy affair, unless you cook the egg enough that it runneth not over everything as soon as you lift the burger to your face and take the first bite. And if you cook it that much, you're kind of defeating the purpose, because it's the egg white AND soft yolk combining with a good quality ground beef that result in an incredible burst of flavor. So ... yes, I liked it, but it was such a pain in the ass to eat that I only had it again a couple of times.
I just realized that I'd get some of that flavor enjoyment in later years by occasionally ordering a hamburger patty, instead of bacon or sausage, with my eggs at L.A. coffee shops. Actually, I probably had it at the Hamlet in those early days. That's a combination I mostly remember from California. You could probably get the eggs "with patty" at a typical northeastern diner, or maybe even a Denny's, but I don't think I've seen it on any menus. But that way, you're eating it on a plate, with a fork, without the bun and other burger toppings that help to explode the damned thing.
