I've read about but not made nor eaten an American hamburger omelette. However, when the dish appears in a currently airing Japanese drama (dorama), A Girl and Three Sweethearts, I noticed that it is not served with bread but in a shallow pool of demi-glace. That's interesting.
Japanese-style hamburgers [and other Asian burgers] are a bit different because the meat is seasoned with a little onion and sake and softer like meatloaf with the addition of bread crumbs, milk, and egg. The Japanese burger steaks are usually served with a sauce made with a combination of ketchup and wooster sauce.