The worst thing you can do to soon-to-be-boiled egg is start it in cold water. Bringing your eggs from fridge cold to a boiling hot with the water causes the membrane to fuse to the white, resulting in marred, scarred-looking deviled eggs. (And no, I’ve never found the vinegar trick to be helpful.)
I like starting my (poked) eggs in boiling water, letting them boil for 10 minutes, then plunging them into an ice bath (if there are a whole bunch) or rinsing them under cold water (if there are only a few and I plan to eat them immediately). I rarely have trouble with peeling. I’d say I encounter one slightly un-peelable egg every dozen or so, which is allowable, considering that eggs and their shells are made up of complex organic systems.
You can also steam your eggs to similar effects (and very tender whites). Just fill a pot with about an inch or water, set a steamer basket inside, cover, and bring to a boil over high heat. Add the eggs and cook for 11-12 minutes before cooling in and ice bath or under cold running water.
