I finally ran out of patience. I got tired of stripping off patches of egg with the shell. So I set out to find the best way to boil an egg and tried all the methods I could find with the help of Google.
Not surprisingly, the best method came from the American Egg Board. Here's how they say we should hard cook eggs, with a few minimal modifications. And it works!
1) The Egg Board says the really fresh eggs are harder to peel, so a week to 10 days in the fridge will make them easier to get out of the shell. During this time air gets through the shell and helps separate the membrane.
2) Gently place the eggs in one layer on the bottom of a pan or pot. Put the pan in the sink. Run water into the pan until the water is 1" over the eggs. Hot or cold doesn't matter. Hot water will boil faster (there is a silly legend that cold water will boil faster - it doesn't). Put the pan on a burner and turn it to medium high heat.
3) Let the water come to a vigorous rolling boil. Put the lid on the pan once the water is boiling and move it onto a cold burner. Set a timer for 12 minutes for medium eggs, 15 minutes for large eggs, or 18 minutes for extra large eggs (egg size is on the box).
4) When the time is up put the pan in the sink. Run cold water into the pan with the hot water still in it until it overflows and let it overflow until the water in the pan is cold, about 4 minutes. Turn the water off and let the eggs sit for about 5 minutes until they are cool. This allows them to cool gradually without shocking them. It helps keep the shell from cracking and makes peeling easier.
5) With a pen, mark the boiled eggs with a "B" so you can tell it from the raw eggs. Now put them into the refrigerator until you're ready to use them. Try to use them within a week. If you plan to use them soon after boiling, peel them as soon as they cool because the inside shrinks a bit.
6) To peel a cooled egg, gently tap it on the countertop or table until it has cracks in it. Roll the egg on the counter or between your hands until the cracks turn into small crackles all over the egg.
7) Use your fingers to start peeling off the shell at the large end of the egg where the airspace is. If you need to, you can hold the egg under running cold water or dip it in a bowl of water to make peeling easier. Throw out the pieces of eggshell when the egg is all peeled (my wife puts them in her compost pile).
Click here for my favorite egg salad recipe, Mayo Mojo Eggs.