In a pot, boil enough water to cover eggs.
Gently place eggs into boiling water using a ladle, and boil for 6 minutes. (This will make a very runny yolk.) Swirl a couple of times while boiling to help set yolks near the center.
While the eggs are boiling, pour the mentsuyu and water into a small ziploc bag (sandwich bag size) or tupperware. Also prepare the cold water bath in a bowl with ice cubes.
Using a ladle, take out the eggs from the boiling water and cool them in the cold water bath. Leave them in the ice water for a few minutes. If the water becomes warm, add more ice cubes or change the water to keep the temperature cold.
For an easy peel, very gently crack the eggs all over and put them back in the cold water bath to let the water come in between the shells and eggs.
Peel very slowly and gently. Once the eggs are completely peeled, put them in the mentsuyu bag/tupperware.
Let the eggs marinate in the refrigerator for at least 1 hour before eating. If necessary, turn the eggs in the container for an even color and flavor.