To me, the most challenging thing about a hard-boiled egg is not cooking it, but smoothly peeling it.
While there are many ways to approach this challenge, I recommend not to boil the egg, but to steam it. Steam for 14 minutes, then transfer to an ice-water bath (at least 50:50) for 15 minutes (Kamozawa and Talbot).
The shells almost fall off, leaving a glisteningly smooth surface (click on the image and notice the detail in the window reflection)
With these timings, the yolk is cooked through, without discolouration around the outside of the yolk.
Its a reliable and reproducible method:
So, the term hard-boiled has to go, because they are not boiled. Hard-steamed it is for me from now on…
Eggs can also be steamed in a pressure cooker. This one below was cooked for 5 minutes at full pressure, cooled down rapidly and then put in an ice-bath for 15 minutes. Notice the creamy soft yolk. It is more delicate and a little more difficult to peel.
While steaming works, I have been unable to find out why.
There seems to be a difference if the egg is boiled in water, so I presume it has something to do with the porosity of the shell. Perhaps when boiled, water can migrate into the egg before the white has had a chance to set, hydrating and swelling it so that it binds more strongly to the inner membrane and shell.
The ice-bath period might be a consideration also. If water were to migrate through the shell into the now-cooked egg, it could help separate the egg from its shell. This could be helped if the egg shrinks a little with cooling. Although this is presumably common to both steamed and boiled eggs.