The Not so Hard Cooked Egg
It's the temperature differential: The white of an egg dropped into boiling water cooks much faster than the yolk at the center, and that’s trouble. By the time the yolk sets, the white is tough. And if the egg stays over high heat too long, or isn’t cooled quickly after cooking, sulfur in the white will react with iron in the yolk, creating that nasty off-colored ring. Yuk!
First of all, before you even cook the eggs, if you store the eggs tip-point down, this will center the yolk. Also - use fresh eggs. The older eggs will help in making the green rings. By the way, if an egg floats when put in water, throw it away.
Boiling: Keep the temperature of the egg white and yolk close in temperature. Heat the eggs gradually. Place them in a saucepan, cover them by an inch or two with cold water, and set the pan over high heat. Add a bit of salt into the water before boiling the eggs. This will make the peeling easier. As soon as the water reaches a full boil, remove from the heat, cover the pan, and let the eggs stand for 10 minutes. This cooks them gently and keeps the whites from toughening, and there will be no cracked eggs while boiling. Peel the eggs immediately under cold running water; or, if you’re not wanting to serve them hot, set them in an ice water bath. This lowers the eggs temperature and minimizes the pressure that causes sulfur rings to form.