I never had these when I was a kid, but I did make them for my kids. The Frugal Gourmet suggested the idea in one of his books, so we use his name for them - Eggs Looking at You. My kids liked them, but I think part of it was the novelty.
I like the Kangaroo Egg idea. Might have to try that one of these days.
Originally, when I was young, we cooked eggs in bacon grease and just scooped up oil onto the yolk to put a thin skin on the yolk.
As we got older, smarter, and healthier we did away with the bacon grease and moved to non-stick pans. In order to continue putting a nice cooked skin on the yolk, when the base of the egg is cooked, instead of flipping it, you put around a couple of tablespoons of water in the pan and throw the lid on. Cook for maybe another 30 seconds to a minute.
It takes doing it a few times to get the hang of it. Too much water puffs up the egg, too little and you don’t get the nice white skin on the yolk. When it’s done right it makes a perfect egg with runny yolk, and you use very little fat in the cooking process. Without having to flip the egg, the chances of breaking the yolk are minimal.
That’s more what should be called a “steam-basted” egg. A “steamed egg” usually (at least in my experience) refers to an Asian dish that is almost like a custard or eggs that are cooked in their shell in a steamer (like a soft-boiled or hard-boiled egg.)
My mother never made it, but a diner in my hometown did and that’s where I first had it. Their name for it was, interestingly, a “birdseye”. Half Bird in a Nest, half Bullseye. Interesting.
Their version had cheese.
“Sun toast” and “Birdseye”
I voted Never heard of it, sounds nasty and This poll is broken because … it doesn’t sound nasty, but doesn’t sound great, or even good, either.
I don’t remember where I learned how to do this, but I used to all the time - and I’m going to make one tomorrow for breakfast. The cut out hole fried is good also.
I think I used to butter the top and put butter in the pan for frying, and then flip - but I don’t remember and am going to have to experiment. Sometimes the Dope is worth every penny!
“V For Vendetta” was the first and until this thread only source which i have ever heard of this. thought it was a british thing.
but much like after seeing it in that movie this thread makes me want to experiment and try to make one. with plenty of butter. hmmmmmmmmmm
We rarely had the eggs in toast because most of the time we had what you described but with vegetable oil. My mom would always warn me that the eggs at restaurants was not going to be the same so I never ordered anything but scrambled in restaurants. We called them fried eggs but I have since learned they are basted eggs. And yes, since you don’t flip the eggs you are less likely to break the yolks.
Grew up with these. They were called “Bullseyes” in my family (like a few other people above).
For best results, use cookie cutters to cut out the inside of the bread (heart, star, etc). Then you have interestingly-shaped breadlings to fry on the side 
Never had it, and only heard of it later as an adult. For some reason I thought it was a Brit thing. Or do they do something else tasty with toast & egg?
I would happily have it.
My mother used to make these when I was a kid in the Bronx. I like them, but never seem to make them now. She called them Porterhouse eggs. I don’t know where she got the name; Googling doesn’t seem to turn up anything.
First time I ever had these was on vacation to Disney World and they called them “Goofy Toast”. That name pretty much stuck around our house, though I’d also heard it called Egg in a Basket.
It definitely isn’t in the same class as eggs benedict, but for a quick breaky tailored to please a child or hung-over adult, it is better than it sounds. (A little seasoning on top and you are golden.)
Actually, I have made this for my wife & daughter maybe twice… tonight (due to nostalgia brought on by this thread) I told my wife I was thought it would be nice to start Saturday morning with Egg-in-the-Holes and she suggested that perhaps Eggs Bennie would be better. She is a spoiled, spoiled woman.
Frog in a hole (I know, very similar to toad in a hole, but toads are gross and frogs are cute). 
My mother called it wombat in a hole, Australia in the 1970s.
Oh I know you don’t but it is presented as a valid choice!
My mother used to make this for me for dinner when I was little…never gave it a name. When I was in home ec. Bergen County, NJ, my teacher, Mrs. Saunders, called them framed eggs…we made them when we were studying, naturally, eggs, in Foods I
I saw something similar a few weeks ago, but it added bacon, cheese, and extra bread and egg.
(Kicking myself for not bookmarking that one.)
Another one worth trying:
Cook bacon halfway. Line a muffin tin with it. Add an egg and bake until done.