Start by bringing a small pot of water to a boil.
Carefully place the eggs into the simmering water once it's boiling.
Let the eggs boil for about 8 to 9 minutes.
Immediately transfer the boiled eggs to a bowl of cold ice water and let them cool for 5 to 10 minutes.
Once cooled, crack and peel the eggshells.
Separate the egg yolks from the whites. Chop the egg whites into small, diced pieces.
In a bowl, mix together the egg yolks, Japanese mayo, salt, black pepper, sugar, mustard, and cream or milk until you get a thick sauce or paste.
Add the diced egg whites to the bowl and combine everything.
Prepare the bread by removing the crusts and spreading butter on both sides.
Layer thinly sliced cucumber on one side of the bread, if desired, and then add the egg filling on top.
Seal the sandwich with the other piece of bread and cut it in half to enjoy!