If you ever make hard-boiled eggs, here’s a tip you might want to try. Bring the water to a simmer, THEN add the raw eggs and proceed as usual. Adding the cold eggs to the boiling water results in eggs that are easier to peel. I’ve done several dozen eggs of different sizes and freshness since I heard of this tip, and the ease of peeling – along with the egg peeling cleanly – is greatly improved.
Some say very fresh eggs don’t peel as neatly or cleanly, but with this technique, the freshest eggs I boiled peeled easily.
Also worth doing – poke a little hole in the big end of the raw egg with something like a pushpin before putting it in the simmering water – it decreases the chance of the egg cracking while cooking.