Even farmers advise letting eggs eggs grow old before trying to hard-boil them.
The reason is, as eggs get old they deteriorate. The egg's attachment to the inside of the shell grows weaker, making it easier to peel when hard-boiled. The membrane that holds the yolk in the center of the white also breaks down, which is why yolks sometimes end up in the strangest places when you hard-boil eggs.
However, there is a method for cooking new eggs that solves the peeling problem. And wouldn't you rather have a fresh egg in your salade Nicoise?
As it happens, I am cooking a dozen old eggs that we found one day on our doorstep. We still don't know who is leaving us eggs. But since they were beyond the "use by" date, I'm hard-boiling them.
The trick is to move the eggs back and forth bettween boiling water and ice water. This causes expansion and contraction between the shell and the egg inside, making peeling a breeze. The method is described in detail in Julia Child's The Way to Cook.
For a dozen eggs, place the eggs in the bottom of a tall stock pot and cover with 3 1/2 quarts of cold water. Over high heat, bring the water to a boil and immediately remove the pot from the heat. Cover the pot and let it sit for 17 minutes. Remove the eggs with a slotted spoon and transfer them to a large bowl of ice water.
While the eggs are sitting in the ice water, put the stock pot back on high heat and bring the water back to a boil. When it is boiling, transfer the eggs back to the pot for 10 seconds. Then move them back to the ice water.
When the eggs have cooled, you can crack the shells and peel them. The eggs should be perfectly cooked (none of that grey ring around the yolk) and easily peeled. Deviled eggs never looked better.