Ok. So I've tried the following twice, resulting in the easiest peeling eggs that I've ever known in a lifetime of hard boiling thousands of eggs. I have no idea why it works, it's certainly not that unusual. The last time I tried this, yesterday, I made a point of using eggs that I'd purchased just the day before. My experience had always been that older eggs were easier to peel once boiled. These eggs were fresh, with almost no air pocket in them and the shells came off perfectly in just seconds.
1. Start with room temperature eggs. I just leave them out overnight.
2. Bring water to a rolling boil.
3. Drop the eggs (gently) into the water and boil them hard for exactly 3 minutes.
4. Lower the heat and simmer the eggs for 12 minutes.*
5. At the end of the time period, plunge the eggs into ice water and leave them sit at least 30 minutes.
* The 12 minute simmer can be adjusted, as needed. I'm at 5000 ft elevation, where water boils at just 203 degrees, so you might use 11 minutes at sea level. You want to avoid overcooking the eggs, resulting in green-tinged, sulfur-smelling yolks.
Don't use cold eggs, as it affects the cooking time and the eggs tend to crack when subjected to boiling water.
The crazy thing about this is that it's very similar to how I've been hard boiling eggs for 20 years. Usually I would place them in water, bring them up to a boil for 1 minute, turn the heat off, cover the pot and let them sit for 15 minutes or so. I think the key may be the initial 3 minute boil.