Here I incorporate my own experience which includes teachings from many other sources including my mother, Alton Brown, America's Test Kitchen and others.
White: contains hydrogen sulfide
+ too much heat and too long, will begin to smell of "rotten egg" (sulfur)
Yolk: contains iron
+ enclosed space (boiled in shell), will form grey-to-green around yolk (when hydrogen sulfide is forced into yolk meeting iron resulting in iron sulfide)
Use butter, cook only until just done, cook at medium-low to medium only sunny-side up, over easy (problem won't occur) over-medium, over hard (lower heat dramatically, take more time reaching that phase or consider smashing the yolk by pressing down on egg just after turning).
Don't use very fresh eggs: age weakens the membrane between the egg and shell. A couple of weeks in the refrigerator should do it.
Start cold, bring to a boil, remove from heat, wait 11-15 minutes, then plunge into ice water. Once cooled, crack all around and put back into fresh ice water to release shell from egg.
If you're using the egg in a salad or other application where you really need "hard," wait 15 minutes (instead of 11).
Peel immediately: that membrane between the egg and its shell will begin to rebond as the egg cools and time passes.
Start cold, bring to a boil, remove from heat, wait 2-5 minutes, then plunge into ice water or serve immediately.
For eggs mimosa, deviled eggs, potato salad and other uses, it’s important to hard-boil eggs, then shell them without damage.
Who knows the best way to do this stuff? I have always had a struggle doing it well, but I find it much easier now. Here is what I do.
Hard-boil only the freshest possible eggs. Place them in a large amount of cold water with salt on the burner. Raise them to a rolling boil. Turn off the burner, lid the pan and set a timer for 10 minutes.
After 10 minutes, empty the hot water from the pan and run cold tap water into it until the eggs are cold to the touch (they will still be warm inside). Move the pan to a work surface and add ice cubes (if you want; I haven’t decided if it’s really helpful to do this yet.)
You may think that you can set the eggs aside, cold, and do this any time. I have found that the quicker you shell the eggs, the easier the shell comes off without trashing the egg. However, another trick is to crack the shell all around, then plunge back in ice water and wait a bit.
Dry each egg with a towel just before peeling. Crack the egg lightly all over by tapping against the work surface. Find a weak spot and start peeling by lifting egg shell. Work open a large spot and then, if not already broken, carefully pinch and break the inner membrane. If the surface of the egg is shiny and slippery, you have succeeded and this will a) make peeling easier and b) not leave the slightly nasty membrane for consumption. Finish peeling using the side of the last two joints of your index finger taking care to stress the egg as little as possible.
Like me you want to boil the eggs, then set them aside in the refrigerator for use tomorrow? Don’t. Complete them now including peeling or you will lose most of them to peeling damage. After they’re peeled, rinse and drain them in a colander. Spread a paper towel folded in half on the counter and blot each one before putting it into a zip-lock bag. Store bag in a suitably sized bowl in the refrigerator. The eggs will keep at least one day, maybe more.
From America’s Test Kitchen, to ensure that the shell of a hard-cooked egg peels off without leaving chips or gouges, use this technique. Before cooking, prick a tiny air hole in the wider end of each egg with a tack or push pin. Remove the eggs from the hot water with a slotted spoon and plunge them into ice water. When chilled thoroughly, the shells should peel right off.
The pin prick is new, but the temperature difference aligns with other advice I’ve already included.
(Alton Brown has a whole episode dedicated to eggs entitle, The Egg Files that is worth watching.) The important thing about scrambled eggs is that if you finish scrambling them in the pan, cooking them solid and dry—what passes for done among people, then by the time they reach the table, they’ll be the usual uninteresting, over-cooked underdelight millions of American are used to consuming in place of food every day.
Editorial rant: The reason Americans put ketchup on eggs and hashbrowns is because they are so used to eating pig swill and it needs ketchup to be palatable.
We eat what amounts to pig swill because of unenlightened and uninterested cooks. We become so used to bad cooking that we go eat scrambled and dried egg folded up on a muffin at MacDonald’s and think it’s food.
Crucial tips for scrambled eggs
To make the scrambled eggs commonly available in the hotels of business travelers, add a little cream before scrambling. Scramble in a non-stick pan over medium heat until the egg begins to seize, then turn heat down and carefully fold cooked and liquid over and over slowly seizing it until it a great deal of it is seizing. Remove and chaffer immediately. It will continue to seize in the heat it holds, cream will balance that, however, by ensuring that it will not dry out. Alton Brown has a rule of thumb for the ratio of any dairy you add to scrambled eggs, one tablespoon of buttermilk, cream, etc. per egg.
It will take you two or three tries beforing determining just how long to chase scrambled egg around the pan before serving. If you serve egg that is solid, custardy bits floating in water sweated out of the egg, they’re nasty: don’t cook so long and add a bit more cream. The cream may enrich with flavor, but its main purpose is to keep the eggs more moist after they leave the burner.
Eggs and hashbrowns make for a heavenly breakfast. If they are cooked carefully and with the cook paying attention to what it means to cook them right, there will be no temptation to put ketchup on them except out of habit.
Usually, we serve eggs on the plate in several different degrees of doneness: sunny-side up, over-easy, over-medium and, rarely, over-hard or well-done.
First note, as quoted in USDA statistics by someone in the episode of Alton Brown’s Good Eats already mentioned, that the risk of salmonella in an egg is a) minimal and b), unlikely to be bad enough to make you sick unless you eat a lot more than one egg and unless your immune system is faulty.
Sunny-side up can be done so the the yolk is set to varying degrees by the use of a lid on the pan. Cooking on lower heat with the lid on is key. There's a more exact method below that does 4 eggs at a time.
Over-easy is best done by one egg or two, never more, in a small skillet with lots of butter over low heat. Once the white is set enough to hold the egg together during the flip, it’s ready to turn. Use the same flipping movement employed when tossing vegetables you are sautéeing in a pan. Also, you can’t flip the egg, obviously, if even the smallest part of it is stuck to the pan: use a non-stick pan, enough butter and use a spatuala to disengage it before attempting the flip.
If you are doing over-easy on the griddle instead of in a skillet, you must wait until the white is set a little longer before turning with your spatula.
Once the egg is flipped, count ten seconds then remove to serve. Beyond ten seconds, you are reaching for over-medium, then over-hard.
An alternative to the above for squeamish children, for a sandwich or for times you are eating without toast or hashbrowns and don’t want to scoop up liquid yolk from your plate: after the turn, squash the egg to splatter the yolk and distribute it evenly across the white. Don’t think you have wait for the yolk to cook; it’s already done enough not to run after a couple of seconds.
...that don't suffer from snotty whites (shudder). People who like "over-easy" really prefer "sunny-side up," but don't like under-done whites. The secret is a) to let the whites spread out thinly and b) to cover the skillet to get the whites all the way cooked. This method also imparts just a hint of crisp to (only) the edge of the white.
Perfection: whites just done, yolk warm yet runny. The secret is that plunging eggs into boiling water suffer from broken egg shells and uncertain heat. Steam is a constant 212° or better (or it won't be steam), so it's easier to predict cooking time. Only using ½" water means it's mostly steam doing the job—that's the secret!
Here are some things that will save your soufflet.