Water vs. milk/cream (or nothing) in traditional (French) omelets

Question

Previous questions here have addressed the reasons why water or milk is added to scrambled egg mixtures in cooking. However, the two questions I've linked seem to parallel a distinction I've noticed sometimes in recipes or in instructions given by chefs and often repeated on cooking forums: one can add milk or cream to scrambled eggs, while only water is appropriate for omelets, particularly traditional "French" omelet styles.

Some chefs claim that using milk in traditional omelets will make them "tough" (or even "watery"), so they recommend only a small amount of water. (Some traditionalists, of course, also discourage any additions to the eggs before cooking for a French omelet, even leaving any seasonings or herbs until the very end.) On the other hand, I have seen other well-known chefs encourage the addition of milk or even cream to omelets to increase richness.

In my own cooking, I have found that adding any type of liquid can increase the lightness and "fluffiness" of the final omelet, and for that reason it's often worth the extra few seconds it adds in cooking time (which can risk toughness). But I've never noticed a significant difference in toughness when adding milk compared to water, nor have I noticed a significant increase in richness (which I find is often determined more by the amount of butter I put in the pan). In fact, the prohibition against cream seems a bit odd to me, given that many chefs also advocate a rather large amount of butter for omelets which often ends up being stirred into the eggs while in the pan.

So -- my question: is there actually any science behind the claim that one should only add water to a French omelet, since milk (and possibly cream as well) could toughen it? Or is this just another culinary myth? Also, is there any other scientific reason to withhold water or milk from a traditional omelet, other than very slightly increasing the richness of the egg flavor? (Keep in mind that the amount of liquid I'm talking about here is very small, usually no more than a tablespoon for 2-3 eggs.)

(Just to be clear, a French omelet is generally cooked quite fast in a hot pan, often stirred and/or shaken rapidly during cooking to raise the egg temperatures as fast as possible, and then folded and unloaded from the pan while the outside is pale to golden yellow -- never brown -- and left creamy ("underdone") inside. I've occasionally seen chefs advocate a slower approach, but the final product is always the same: barely colored on the outside, creamy to slightly runny on the inside, and the overall goal is to maintain maximum tenderness, often with little or no filling at all. This is opposed to a more "American" or "country" style omelet which can be tougher and perhaps somewhat browned to create a more durable shell used to hold a large amount of heavy fillings.)

Answer 1

Americas test kitchen has tested this in a prior issue. I cannot put my hands on the issue right now which explained their process and the results. It also contained a good deal of explanation about the science of the process. They tried all sorts of liquids in their omelette and scrambled eggs recipe. They found that water did make the eggs slightly fluffy but did not slow the coagulation process. This led to a tougher scramble than milk. They wanted the same result for an omelette with a denser texture. The end result being that butter was better for an omelet.

However, some of the result was later reported in a Smithsonian Magazine Article. It is a much more compact summary but fairly on the nose for your question.

Their answer was as follows:

Add milk to scrambled eggs, frozen butter to omelets: If you want scrambled eggs, most of us know to throw in a bit of milk or butter while scrambling. That’s because the lipids in the dairy coat the proteins in the egg (11 percent in the whites and 16 percent in the yolks) and slow down the process of coagulation, a.k.a. when the proteins are denatured and unfurl, releasing much of the water in the mixture. Adding fat helps keep some moisture in and fluff up the final product. But the same does not go for omelets. “While scrambled eggs should be fluffy, an omelet is more compact,” the authors write. While milk works for scrambled eggs, it can add to much moisture to an omelet. The chefs recommend frozen bits of butter instead, which melt more slowly and disperse more evenly. And it turns out you can go ahead and salt the eggs before you even cook them up. Because salt affects the electrical charge on the proteins, it weakens the bonds between them, preventing overcoagulation. Bring that up at your next brunch.

The Science of Good Cooking: Tips From America’s Test Kitchen