Keep the shell under water while peeling, running faucet or pan. Peel it while still warm, not hot or cold, and start at the large end where the air pocket is. The water lubricates the membrane so it doesn't stick to the albumen (egg white). Always cool hot eggs as soon as possible while keeping them submerged, to avoid oxidation of the sulfurous layer on the egg yolk, which turns green. Cooked egg whites are oxygen permeable.
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1. First take your egg then hit in on a hard surface so you get a nice crack in it.
2. Put your nail under the crack and begin to pull away the shell. There should be a shell layer then under that there should be a "skin" type layer. You want to get under the skin layer.
3. I find it easier to rotate the egg while I'm shelling it.