Fill a medium saucepan with water and bring to a boil over high heat. Reduce the heat to medium so that the water is simmering just below the boil. Gently lower the eggs into the water using a slotted spoon. 

Let the eggs simmer for 6 ½ minutes. 

While the eggs are simmering, place several ice cubes in a bowl large enough to hold all of the eggs. Add cold water to cover. 

When the eggs are finished cooking, pour off the hot water and run cold water of them for a few minutes to cool them down. Transfer the eggs to the ice bath and let them sit until they are completely cooled, about 5 minutes. Drain.

Carefully peel the eggs.

In a medium saucepan, combine the soy sauce, sake, mirin, water, sugar, and ginger and bring to a boil over high heat. Reduce the heat to medium-low and simmer, stirring frequently, until the sugar dissolves completely, about 3 minutes.

Transfer the mixture to a jar or container large enough to hold both the marinade and the eggs and let cool to room temperature, about 20 minutes.

Add the peeled eggs to the soy sauce mixture, cover, and refrigerate for about 8 hours (it is not ideal to leave them longer than 12 hours as the texture of the eggs may begin to suffer).

Remove the eggs from the marinade and store them, covered, in a bowl in the refrigerator for up to 3 days.