First, place eggs in a saucepot large enough to hold them in a single layer.
Fill the pot with cold water to about an inch above the eggs.
Cook them over medium-high heat just until boiling.
As soon as eggs come to a full boil, remove from heat immediately and cover with a tight-fitting lid. Set timer and time for 15 minutes.
When time is up, strain off the water and cover with cold tap water. At this point, I add a generous amount of ice and let the eggs rest until chilled.
Once chilled, tap eggs on the counter at both ends and roll back and forth across countertop to crush shells completely. (Don't roll extremely hard or you'll break the egg apart.)
Once cracked, begin peeling at the large end, under cool running water to help loosen the egg from the shell.
Medium and soft-cooked eggs following the exact same method, just adjust the time they stay covered in the hot water.