Scrambled eggs are physically a gel, as are other cooked eggs. Raw egg proteins are very large molecules but they are normally folded in on themselves to form smaller lumpy balls that are separate and float in the egg’s water (eggs are 90% water). Agitation by heat unfolds these proteins. When unfolded, the proteins are large and ungainly and they entangle and bond to each other. This process creates a random mesh that traps the water (and fat) in the egg and thereby sets it. The process is irreversible (unlike a gelatine-set gel for example).
However, if the eggs are heated further, the mesh gets tighter and instead of holding water, the water gets squeezed out (syneresis). This leads to dry scrambled eggs sitting in a puddle of water.
The transition between a perfect gel and a dry one can be quite quick. It is for this reason that classic scambled eggs are cooked in a double-boiler (a bowl over simmering water but not in contact with it). It limits temperature and adds leeway, but it’s a bit of bother.
To delay the transition in a skillet (or with the double-boiler method too), add some fat. Any dairy fat will do (butter, full-cream milk, cream or a combination). The additional fat will get in the way of gellation and slow the process down, giving a greater time-window to get the texture just right. I find the easiest method is to melt butter in the pan (be generous, it’s not that bad for you) then add the blended eggs (with or withought milk/cream) and stir furiously at first to mix with the butter. Then cook them slowly with a bit of languid stirring. Take the eggs off just before they reach the desired texture, as they will continue to cook a little in their own heat – this is still a tricky call but the fat will help.
Whether to add salt or not tends to be controversial. The hypothetical case against is that it draws water out, although out from what and into where isn’t clear (the sodium and chloride ions will be dissolved in the water trapped in the mesh and uniformly distributed in the gel). In fact, salt’s main effect is to interfere with protein-protein interactions and thus slow gellation to our advantage, although it is only a minor effect. In any case, pre-salting is fine and a more uniform way to season the eggs than at the table.
I imagine you will be expecting me to say that if you want the eggs perfectly cooked at their gellation temperature then sous vide is the answer. Correct. Although, you might also be surprised to know that I prefer the skillet method for texture (sous vide they are more custardy). Egg yolk proteins gel at around 70C, and the two main proteins in the white (ovotransferrin and albumen) set at 65C and 80C respectively. As a rule, a typical blend of yolk and white will gel at around 73C. A suggested starting point for sous vide is 72C for 30m. Try a higher or lower temperature according to taste, but keep the 30m – it needs about that time to reach thermal equilibrium. Don’t forget the fat. Perhaps massage the bag occassionally.
It’s a fine line between scrambled eggs and an omelet (the French do a sort of scrambled egg omelet) and the above considerations apply to omelettes as well. There are modernist approaches to omelettes, but I’m not sure they contribute all that much to home cooking (but here’s an example). And here’s the master, Jaques Peppin, showing how to make two types of omelet the time-honoured way (if only I could do it this effortlessly). The plating method works a treat (much of the time).