More like hamburger roll/sliced bread buttered + Fried egg + fried bacon + sauce (tomato or BBQ)
The American version might be a little drier (sauce isn’t common, and butter’s not universal), but a basic roll with fried (or scrambled) eggs and bacon/sausage as a takeaway (or foodstand) breakfast is pretty ubiquitously available. The bacon/sausage is optional, the egg is not (well, people eat hot buttered rolls too), but I’m getting at the fact that you won’t generally see the equivalent of a U.K./Ireland butty/bap with bacon <only> – not sure if they have those in Australia.
I’ve seen that done with bagels, English muffins, and (American) biscuits. Occasionally you might see it done with toast, like at, say Sonic*. But I don’t think I’ve ever seen it with a hamburger bun.
Oh, and the sauce is usually optional, and is more likely to be (American) ketchup or even salsa. I personally like it better without any sauce.
*Site contains flash, so I linked you to the menu nutrition page, which does not.
My contribution: I found out the reason my mom’s spinach tastes so good. She drops an egg in it, like you would Chinese fried rice.
There’s also loco moco: http://www.hawaiimagazine.com/blogs/hawaii_today/2009/5/20/loco_moco_Hawaii (burger on rice topped with fried egg and gravy)
Don’t forget cheese. A slice of American cheese is a common “dressing” on an American breakfast sandwich (“breakfast sandwich” or “egg sandwich” appears to be the generic name for this concoction).
The standard configuration is
Bread product–bagel, English muffin, biscuit, hard roll, toast, Kaiser roll (less common) or even a croissant (though the “croissant” tends to be a greasy, chewy thing under such conditions)
+Egg–scrambled or fried (at chain places, often an egg-based patty)
+Meat–egg, bacon, ham, Canadian bacon, sausage patty
+Cheese
The last two are not mandatory; egg & cheese, or egg & meat (no cheese) sandwiches are common.
I’ve also eaten “breakfast wraps” with scrambled eggs etc. rolled in a flour tortilla, and “breakfast crepes” with eggs etc. rolled in crepes (delicious if you can get them).
Similar idea with grits, which is what I’d prefer.
A bowl of ramen noodles.
If my dad gets roped into making breakfast, it’s invariably his Sexton Special: Buttered toast topped with hash topped with eggs over easy topped with a slice of cheese.
Sooooo yummy.
Chef Guy and Pulykamell: It would be my pleasure to dine with you! I love my oatmeal savory and have put the following in my oatmeal: eggs, mushrooms, veggie burgers, potatoes, cheese, onions, curry powder, hot sauce…mmm…
Chile poblano. There’s a Mexican place near here that serves chile poblano with a fried egg on top. Delicious!
Yes yes! I had it at a pub once, and then decided I had to make it at home.
It was good at home, but, much like sushi, well worth paying someone ELSE to do it for me. 
Or beat a raw egg and stir it into your steaming hot grits after you take the pan off the burner. Yum.
fried egg sandwich - over easy egg with yolk just a little runny, 2 slices toasted bread and yellow mustard.
yum!
How about: “Lobster Thermidor a Crevette with a mornay sauce served in a Provencale manner with shallots and aubergines garnished with truffle pate, brandy and with a fried egg on top and spam.”?
A fried egg on top of roasted or home-fried yam pieces. Good stuff.
Ever since I heard the ending theme song of “Frasier,” I’ve thought that scrambled eggs would be awesome on a salad!
A favourite snack of mine is to boil up a bowlful of garden peas and add two soft-poached eggs on top, with salt and pepper.
Helps if you’re a pea-fan of course.
Has nobody said refried beans yet? Having leftover beans and rice is a great event chez moi, cause it means huevos rancheros is coming soon.
mmmmm waffle sandwich…
1 crisp platesized waffle with butter, 1-2 fried eggs, 2-4 strips crunchy bacon 
the waffle catches all the crumbs and egg bits. Om nom nom! now I want one.
I think it’s a great idea, but folks are routinely surprised by it. Who needs syrup?
I’ve eaten the egg on pizza in France and I’d love to try it at home. “A cheval” I think they call it [on horseback].
Does anyone know when in the baking process to put the egg on the pizza? I’m afraid to wind up with raw egg all over a pizza.