Heat water in a medium or large saucepan until it's just starting to boil (a full rolling boil can sometimes cause the eggs to crack). You want enough water to completely submerge the eggs by at least 1 inch.
Once the water is boiling, remove the eggs from the refrigerator and use a slotted spoon or ladle to gently place them in the pot.
Reduce the heat to medium so the water continues to simmer without smacking the eggs around, and cook for exactly 7 minutes (you can cook them for 8-9 minutes if you want a firmer yolk).
While the eggs are cooking, combine the soy sauce, mirin, and water in a large resealable plastic bag.
After 7 minutes, transfer the eggs to an ice bath to halt the cooking process. Allow them to cool for around 3 minutes, then peel (these will be softer than hard-boiled eggs, so peel carefully).
Place the eggs in the prepared marinade and seal the bag. Place in the refrigerator for at least 4 hours or up to 12 hours.
Remove the eggs from the marinade, slice in half and place in ramen soup. I also love serving them over rice with a drizzle of toasted sesame oil, some scallions, and a sprinkle of furikake (a Japanese seasoning).
You can use more or less water to adjust the intensity of the flavor. See post notes for some different ways to reheat the eggs.Adapted from Just One Cookbook.