Ajitsuke Tamago (Marinated Soft-Boiled Eggs for Ramen)

In a medium bowl, whisk together water, sake, soy, mirin, and sugar until sugar is dissolved. Set aside.

In a medium saucepan, bring 2 quarts of water to a boil over high heat. Pierce the fat end of each egg with a thumbtack to make a tiny hole. (This prevents them from cracking and eliminates the air bubble at the end.) Using a spider or slotted spoon, carefully lower eggs into water. Reduce heat to maintain a bare simmer (about 190°F/88°C). Cook for exactly 6 minutes for a liquid egg yolk and 7 minutes for jammy egg yolk. Drain hot water and carefully peel eggs under cold running water. (The whites will be quite delicate.)

Transfer eggs to a bowl that just barely fits them all. Pour marinade on top until eggs are covered or just floating. Place a double layer of paper towels on top and press down until completely saturated in liquid to help keep eggs submerged and marinating evenly. Refrigerate and marinate at least 4 hours and up to 12 hours. Discard marinade. Store eggs in a sealed container in the fridge for up to 3 days. Reheat in ramen soup to serve.