Select a saucepan that's wide enough to fit all your eggs in a single layer. Fill the saucepan with enough water to cover your eggs by about 1 inch. Place saucepan on the stove and add a glug of vinegar and a couple big pinches of salt. Bring water to a boil.
Working quickly but carefully, very, very gently lower your eggs, 2-3 at a time, into the boiling water. I use a small handheld strainer. Be careful not to crack them. As soon as all eggs are in the boiling water, start your timer. See Note 2 for timing recommendations.
When time is up, carefully remove the eggs from the boiling water and transfer to an ice water bath using a slotted spoon or small handheld strainer. Alternately, quickly but carefully drain the hot water from the saucepan and fill with enough cold water to just carefully cover the boiled eggs. Add lots of ice, about 2-3 cups.
Let sit until chilled, about 5 minutes.
Drain water from the water bath and aggressively shake and jiggle the saucepan or bowl, banging eggs up against one another and the sides of the bowl or pan. Toss the eggs 1/2-1" up in the bowl to liberally crack the shells. They should begin to feel almost like mesh and should not have any large intact pieces.
Under running water, peel the boiled eggs. The running water is not necessary to make these perfect boiled eggs so easy to peel but is helpful in washing away remaining flecks of shell.