I prefer my eggs hard boiled. But sometimes I end up with boiled eggs that have runny yolks, either due to my own carelessness or the overeager nature of the house help. Is it okay to boil eggs after they have been completely peeled? Will the results be close to a normal hard boiled eggs? Will I face any issues like the egg opening up and leaching the liquid parts into the boiling water?
I poach eggs, remove and chill them in an ice water bath to stop the cooking and then 'reheat' them in the simmering/boiling water right before getting ready to nap them with hollandaise and then serving them. (to aid in creating mass quantities of eggs benedict to be served at the same time)
I suspect that you can further cook your under-cooked hardboiled eggs with no adverse effects (even AFTER peeling) in the simmering/boiling water. Hope this helps.
I believe that cooking unpeeled eggs would be fine, as I found multiple sources, including here on Cooking SE, which indicate additional boiling to an unpeeled egg is fine. However, reboiling any peeled egg(s) is unlikely to turn out as you are hoping.
The reason boiling them, which is a rather violent process in itself, works so well initially is that the shell is there to protect the relatively delicate egg matter inside. Without that protective shell during the second boiling process, I believe the egg would begin tearing apart under the rigorous movement of the water. In my experience, boiled eggs are easily chopped, broken open, or even macerated.
My suggestion would be that next time, you could peel and check one of the eggs in the batch for done-ness, and if the are undercooked, the others could be finished while still in their shell.