What is the term for serving a soft-cooked fried egg that breaks when the meal is consumed?

There actually doesn't seem to be a special word for this as a technique - as common as it is (to add an egg to the top of a dish and use the runny yolk as sauce), it is usually just the dish name "topped with an egg" or "topped with a fried egg". Sometimes the egg is fried, occasioanlly poached, sometimes it is cracked on top of the dish and cooked in the residual heat - the goal of having the yolk serve as sauce remains the same, though.

Some dish variations specify the egg on top - in some south American countries, a steak topped with an egg is called a "horse-riding" or riding steak. I have heard of pancakes topped with a fried egg being called a "texas one-eye", though i can't find a citation just now. Pasta with a raw egg added, which is supposed to be cooked in the residual heat and make a sauce of the yolk - is pasta "carbonara".

Its a little funny, but as common as it is to find a dish with topped with an egg, so the yolk will double as sauce - it doesn't seem to have a name more specific. (And it is really common, all sorts of cultures have equivalents). You can say "topped with an egg", or use ElendilTheTall or ESultanik's suggestions for how to specify the soft-cooked nature of the fried egg, and people will probably understand you.

Edit: with the help of some comments from Andrew Mattson, Relaxed, and Chuu, we now have a potential term: steak "Au Cheval" (French) or beef "a cavalo" (south American) means a steak with an egg on top. The words look similar-ish enough to seem related in word appearance and meaning, yet from two different countries - which translate the term to horse riding or horseback steak, meaning the egg is on top of the dish. So with that commonality, we might take the term "Au Cheval" and take it to mean when an egg is laid on top of, or "riding" the dish. As a term it won't be too well known right now, someone may not be understood if you just ask for a sandwich or veggie hash "au cheval"... but if enough people see this definition, or start using it, it may become the official term for the cooking technique!

In the US, where people care deeply about the exact consistency of their breakfast/brunch foodstuffs, the term is either over easy or sunny side up. Both denote a cooked white and a runny yolk. Over easy involves flipping the egg carefully to cook both sides, whereas sunny side up involves flicking the cooking oil onto the top of the egg to cook the top. The latter makes it somewhat easier to maintain a runny yolk.

ElendilTheTall beat me to it; read their answer for more info.

I will add that, at least in the US, whenever a menu states that a dish is topped with a "fried egg," it almost always implies that the egg will have a runny yolk. You often see this on sandwiches (particularly hamburgers), and often as an optional addition. Typically, the egg will be shallow fried similar to what is sometimes called the "Spanish Method." This produces a solid white, runny yolk, and crispy browned edges of the whites. Ordering eggs "over easy," "sunny side up," or "soft poached" at most breakfast/brunch places will produce a solid white and runny yolk, but it will typically not result in the crispy edges.

Perhaps the best soft cooked egg to stir into a dish is the Japanese style "onsen egg", an egg traditionally cooked in a hot spring. This can be done perfectly every time at home cooking the egg in a sous vide bath of water at 168 F for twelve minutes then cracked over the food. A perfect poached egg drops from the shell.