Bring a pot of water large enough to hold the eggs in a single layer to a full rolling boil. There should be enough water to cover the eggs by 1 inch. As soon as you set the pot of water on the stove, take your eggs out of the refrigerator. Once the water is boiling, add the eggs, one at a time using a slotted spoon. Set the timer for exactly 6½ minutes.
Prepare a bowl large enough to hold all the eggs with ice water and have it standing by. Once the eggs have cooked for 6½ minutes, remove them with a slotted spoon and place directly into the ice water. Let sit for 5 minutes. Peel the eggs. Place them in a deep container or resealable food storage bag.
Combine the tamari and mirin and pour over the eggs. Marinate in the refrigerator for 3 hours or up to 3 days. If planning to marinate for more than 1 day, add ½ cup of water to the tamari/mirin mixture.
I do not recommend reusing the marinade to make more ramen eggs however, if you plan to make ramen, it can be poured into the broth before bringing to a boil and simmering.