I was going to, because I love a bean-and-egg burrito. But I couldn’t decide if that fit in with the spirit of the OP: is it a Bean Burrito that is made better with an egg on (in) it, or is it a breakfast burrito that is made better with refried beans on (in) it?
In any case, inspired by the OP, I have eaten Enchiladas Montadas (with, naturally, refried beans) each of the last two nights. Last night was leftovers, which consisted of two fried eggs on top of a small stack of cheese enchiladas on top of refried beans.
I can attest to a hamburger with a fried egg on it. Some call them Cajun Burgers, Paula Deen writes about it in one of her books, but calls it something else. It is really good even though it sounds gross. The Cajun Burger normally has everything you’d normally put on a burger, as well as an onion ring, ham or bacon, and hog head cheese. I haven’t tried it with Hog Head cheese so I cannot give an accurate thumbs up or down on that one.
One of my favorite pizzas is the spinach/goat cheese/egg calzone. The local (in the Netherlands) turkish pizzarias usually have them. Yummy.
Also, most of the more chinese/indonesian takaways over here do a nasi (stirfried rice with bacon/pork, veggies and scrambled egg) with a fried egg on top.
This is all conjecture based on various newspaper/magazine and food blogs: If you have an insanely hot pizza over (at least 700 degrees Fahrenheit I’d guess) you can put the egg on top just as the pizza is pulled from the oven. The residual heat should solidify the white and warm the yolk.
But if you’re using a normal home oven, put the egg on a few minutes before you would normally take the pizza out, to get the cooking process started. Unless you’re willing to burn your mouth, pull the pizza before the egg is entirely done; it’ll finish as the pizza cools to a safe temp.
When I was in France and experienced this, it looked pretty much like they pulled the pizza out of the oven, cracked an egg in the middle, and brought it to the table. The yolk was broken, whether deliberately or not I don’t know, so it was cooking and sizzling too.