Begin by bringing a pot of water to a rolling boil over medium-high heat. Gently lower the eggs into the water for medium soft boiled eggs, cook for 7 minutes; for hard boiled eggs, cook for 10 minutes. Transfer eggs to an ice bath to cool for about 5 minutes.

Once cooled, carefully peel the eggs under running water. Mash the eggs in a mixing bowl until slightly chunky. Add sugar, salt, pepper, and Japanese mayonnaise. Stir until well combined, adding milk if using hard boiled eggs.

Spread softened unsalted butter on one side of each slice of Japanese milk bread. On the unbuttered side of one slice, spoon the creamy egg salad. Top with another slice of bread, buttered side down, and trim crusts if desired. Cut sandwiches in half diagonally.

Serve immediately or wrap tightly in plastic wrap for up to 2 days in the fridge. Enjoy as a quick meal or a brunch treat.