I was making a small batch of chocolate cake and decided to flavor it with peppermint extract (as a substitute for vanilla extract). Foolish me added the extract directly to the eggs (not beaten). A small portion of the white became opaque, as if it had been cooked. What happened here?
Your egg whites were cooked by the alcohol in the extract. Cooked, in this case, means denaturing, which means unfolding the protein molecules. There are many ways to denature proteins. Acid, such as vinegar, will denature egg proteins, which is why some people suggest adding vinegar to the water to poach an egg (a practice I disagree with, but that's another subject).
Ceviche, as another example, is fish which is cooked (denatured) in acid (such as lime juice). Although heat is not used, the fish is still cooked. (Note: the acid will kill some bacteria but is not as effective as cooking with heat, so be sure to use fresh fish from a source you trust.)
This is similar to how burning means combining with oxygen, so you can burn metal (rust) without heat (most easily by adding water).
The egg whites contain proteins that depend on water to stay soluble. The peppermint extract is no doubt alcohol based. Thus the alcohol denatured the proteins. However there isn't enough alcohol to denature all the egg white, it will just denature the egg white that it first comes into contact with. After that the water in the egg whites dilutes the alcohol.