I learned this late in life too. Better than “Over easy, but then a little more.”
That’s how I mix my eggs into fried rice, Push everything to the side to make a hole, break the egg into it, slowly push it around until solidifies (doesn’t take long at stir-fry temps), mostly separate white and yellow,. Then break it up a bit, and mix it into the rest.
Never heard a specific name, but I’ll do that to an egg sometimes if the yolk breaks when I start frying.
Sounds like an omelette with no toppings. Maybe that would convey how he wants it?
My go-to breakfast sandwich recipe:
- Toasted sourdough English muffin.
- Fry 1 thick-slice hickory-smoked bacon strip along with a sliced baby bella mushroom and minced garlic.
- 1 slice deli American cheese.
- 1 poached egg, using the 2-step microwave egg-poaching method: microwave ½ cup water and a splash of vinegar (to coagulate the egg-white) in a Pyrex measuring cup to boiling, then add the egg and microwave for an additional 35-seconds. This is the best MW method for poaching eggs to perfection.
Place the cheese on the top half of the English muffin as soon as it comes out of the toaster (assures gooey cheese). Cut the cooked bacon into 3 pieces and arrange it as a lattice on top of the cheese. Arrange the fried mushroom slices and garlic on top of the bacon.
As soon as the egg is poached place it atop the mushrooms, and pop the yolk with the tines of a fork so that it oozes into the mushrooms and bottom slice of muffin (pre-popping the yolk also prevents it popping with you bite into it, which can be messy). Carefully press down on the muffin to in order to have everything ooze together before taking your first bite.
An omelet has you mix the white and yolk into a single pale-yellow consistency. In this case the goal is to mix the egg up even less than a scrambled egg. If you ask someone to make you an omelet, it’s going to be even more mixed up than scrambled eggs.
“American hard scrambled” gets you pretty close, and is what I think of as scrambled eggs (I dislike the super fluffy or runny kind):
That’s not quite what your uncle is asking for if he wants it to come out as one piece, but the texture and “marbling” is there.
Aside from being my preference, it’s also the easiest to make by far. No separate dishes, and no talent with the spatula required.
What you have in that picture is what I picture scrambled eggs to be. I mean, every time I scramble eggs, that’s what they look like. I imagine the OP’s uncle didn’t want them mixed up quite so much.
Otherwise I agree, that’s just asking for scrambled eggs.
While I agree, the first Google hit for “scrambled eggs” is this:
Whisk in a separate bowl until completely homogeneous, add milk, and cook only to a goopy consistency.
Not what I want, but it seems to be the style for “gourmet” scrambled eggs.
I mean… Why? 
If you’re going to make the effort, just do something better, like an omelet or frittata. This seems like using high end ingredients and a panini press to make a peanut butter and jelly sandwich.
Okay, I guess different stroke for different folks…
I’d wager that your average diner isn’t going to give you scrambled eggs like that, though.
Don’t ask me to defend it! I guess they’re all trying to be Gordon Ramsey with his nasty, goopy eggs:
Agreed, and my link earlier says as much. Though at the other end of the spectrum, you get scrambled eggs made from a pre-mix, usually coming out homogeneous and fluffy. Any diner worth calling a diner isn’t going to serve that, but it’s the kind of thing you’ll find in the steam tray at your hotel’s continental breakfast.
That’s similar to how I make my own. I put them in the pan and right before the whites are done, I use the spatula to gently break them and the yolk apart, softly fold together and cook until almost done.
Old Navy joke but works with any branch of the military. Making breakfast the new guy watches the old guy mixing up scrambled eggs. He takes a 5 gallon bowl and mixes up a ton of powdered eggs, then cracks a dozen real eggs and tosses them in shells and all. The new guy is totally confused by this but he’s told to wait and see what happens. Before long a sailor picks a large piece of shell out of his eggs and shouts “Hey you cooks! You’re supposed to keep the shells out of the eggs”. But then another guy shouts “Shut up about it! At least were getting real eggs instead of that powdered crap!”.
if you’re in Spain, you can order huevos rotos (lit. broken eggs)
The egg part of that is exactly like they come out when I use my method above.
That looks icky. I’ve seen videos of Jacques Pepin making the same thing, it looks like watery pudding.
I like to cook it this way and then spread i on some nice buttered toast. Delicious.
like a dip? I’m sure it tastes great, but…the texture turns me off.
That’s how I want my eggs, with the consistency of chunky mayonnaise. 
Oh, wow, lots of replies here. Let me just take this as I go…
Hm, the first hit in a Google image search looks kind of close, but it’s clearly not a standard term, because nothing else is even in the same ballpark.
He’s tried that, but it usually results in the cook either not mixing them at all, so the yolk is still mostly localized in the center, or mixing them completely.
That sounds like it’d be on the runny side-- He likes it all cooked hard, after doing… whatever-it’s-called.
Given how many folks in this thread have said they do it this way for sandwiches, that might be a winner.
The irony is, when he’s cooking himself, he’s really good with eggs. He’s told stories of when he was in Boy Scouts, and all the other boys, making their breakfasts, kept on breaking the yolks accidentally, and then he cracked open two perfect eggs to the awe of all of them… and then immediately mixed them up.
And another story, from his Army days: He and the other guy on KP were making some very large amount of pancake batter, and were proud of doing so without getting any shell in. And when the sergeant heard this, he grabbed a double handful of shells out of the trash and crumpled them in.
That looks tasty to me, but it’s runny, and it’s not all one piece.