I have another method. I poke a small hole in the large end of the egg, and set them in the pan with that end up. I add a small amount of salt to the water and then boil as usual.
It’s actually difficult to hold onto the eggs as I peel them, because the membrane becomes just a slippery mess. Incredibly easy to peel and then rinse off.
I don’t, I have an <insert Alton Brown shaking his fist and cursing at me> egg device. Other than I don’t use it to poach, I adore it [mrAru uses it to poach, he never learned to do it properly.]
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This problem plagued me for years. We have our own hens, so I knew exactly how old my eggs were, but still never could get a clean peel, even with weeks-old eggs. Here is what actually works, consistently, even with just-laid eggs.
Bring your water to a rolling boil. Keep the eggs in the fridge until it’s boiling. Then lower each cold egg into the boiling water with a slotted spoon, and cook as usual. I have tried this with eggs that were laid the same day with excellent results.
A bit of vinegar in the water helps many ways. It will attack the shell, thinning it, and allowing easy peeling. IMHO, its the only way to hard boil quail eggs, they’re too tough to shell otherwise. Also, it penetrates slightly, coagulating the outer white sooner, and that prevents the egg contents from expanding and cracking the shell.
I remember the Peter Cottontail cartoon, they made a big deal about peeling the entire egg all in one piece. I could never achieve that, until I began boiling in vinegar. Its not a simple achievement, but I do come closer.
I use an egg steamer similar to the one linked above, that wants you to pierce the big end of the egg and get them into ice water as soon as they’re done. I make sure that the piercing is under the water as they cool. That sucks some water into the space between the egg and the shell and separates the white from the membranes so they peel more easily.