Common kitchen wisdom dictates that hard-boiled eggs must be “shocked” in an ice bath to stop the cooking process and make them easier to peel, but only half of this is true, and the other half can be easily mitigated, meaning you can leave the ice in the freezer.
Plunging hot eggs into a bowl of ice water has absolutely no affect on their peel-ability. I know this because I have recently been cooking and eating a massive number of hard-boiled eggs, and nary an ice cube has been used to accomplish their peeling. The only water temperature that affects eggs and how easy they are to peel is their starting temperature, which should be boiling; starting eggs in cold water and bringing it to a boil bonds the eggs to the membrane, which makes it extremely hard to separate from the shell. Start your boiled eggs in hot water, and you won’t have this problem. (The only exception to this is pressure-cooked eggs, which are also super easy to peel)
Setting your boiled eggs in a bowl of merely cold water, particularly if you’re only cooking a few, is all you need to get them cool enough to handle. Once they’re no longer finger-blisteringly hot, you can peel them like normal without marring the white.
So ice water won’t affect your egg’s peel-ability, but it is true that it stops the cooking process. Luckily, eggs aren’t as delicate as something like asparagus (a vegetable that really benefits from a good shocking), and you can prevent the carryover heat from overcooking your eggs by simply cooking them for a shorter amount of time in the first place.
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The three eggs you see above were all cooked for five minutes under high pressure in my Instant Pot, followed by a manual release. The one on the left was rinsed under cold water until it was cool enough to peel immediately after the cook time elapsed, while the two on the right were transferred to a bowl of cold tap water, along with another four eggs. As you can see, the yolks on the right are a little more set and slightly paler, but there is no dreaded grey ring of sulfur. If that tiny bit of further cooking is unacceptable to you, you can always shave a minute off of your cook time, and utilize the residual heat to “finish” your eggs.
If you’re happy using an ice water bath, then by all means, continue to do so. But don’t expect it to make your eggs any easier to peel, and don’t panic if one day you go to boil some eggs only to discover you are out of ice. You can still make perfectly cooked, perfectly peel-able eggs, even without those frozen cubes of water.