Fill a small pot about halfway with water and bring to a boil.
Carefully lower the eggs into the pot one at a time. (I like to use a small strainer, but you can also use a spoon.)
Set a timer for seven minutes. Try to keep the water at a brisk simmer without jostling the eggs around more than necessary.
While the eggs boil, fill a medium bowl about halfway with lots of ice and cold water.
When the timer rings, take the eggs out right away, place them in the ice water, and let sit until cool, about 15 minutes.
Meanwhile, in a small pot, stir together sake, mirin, soy sauce, and sugar.
Bring to a boil, stirring occasionally.
When it boils, turn down the heat and simmer for one minute.
Off the heat, add one heaping cup of ice to the pot to both dilute and chill the marinade. (Alternatively, you can place the ice in a 4-cup glass measuring cup or nonreactive bowl and pour the marinade overtop.)
Place peeled seven-minute eggs into marinade.
Cover tightly, and refrigerate for about 24 hours, turning once or twice if you think of it.