Create an ice bath in a large bowl. In a medium pot, bring enough water to a rolling boil to cover the eggs. Gently lower the eggs into the boiling water. For medium soft-boiled eggs, cook for 7 minutes; for hard-boiled eggs, cook for 10 minutes. Transfer to the ice bath for 2 minutes, then peel while warm.

In a large bowl, mash the peeled eggs with a fork until smaller than a pea. Mix in sugar, salt, ground black pepper, and Japanese mayonnaise. If using hard-boiled eggs, add milk for moisture. Adjust seasonings to taste.

Spread softened unsalted butter on one side of each bread slice. On the unbuttered side of one slice, evenly distribute the egg salad. Place another slice of bread on top, butter side down. Trim crusts and cut in half. Optionally, garnish with chopped chives.

If not serving immediately, wrap each sandwich in plastic wrap and refrigerate for up to 2 days.