For hard-cooked eggs that are thoroughly cooked but not rubbery, reduce the heat to a simmer after the water first comes to a boil. Immediately cool the cooked eggs, still in the shell, under cold water, to prevent a harmless but unattractive greenish color from developing around the yolk. For easiest peeling, crack the egg at the large end, which has an air pocket, and peel with the side of your thumb rather than fingertips.