One of the nice things about Fatburger in southern California is that it serves a hamburger topped with egg. I can't find that much around here, but I just ate one at Joe's Cable Car. They'll cook the egg to order (I had over easy) and aren't afraid to serve rare burgers. It was really good, although my cholesterol disapproves. Especially when you get to the yolk and it runs all over the bun and the rare patty, mmmmm (sort of like cooked tartare--excuse the contradiction). My wife thinks the whole idea is gross though, so maybe not for everyone. Milkshakes are OK. The banana shake was made with syrup and not real fruit. Onion rings kind of sucked. If you have a sense of irony, the decor is funny. If you don't, it's still worth going.
I think that Carl's Jr. is also offering a breakfast burger with fried egg and hash browns, but I don't know if it's served all day and I do know that the beef won't be nearly as good (like all "Joe's" the Cable Car grinds its own chuck daily---but it doesn't offer a huge menu like the others, basically just burgers).