Bring a small pot of water to a boil.
Add the eggs gently to the bottom of the pot once the water is simmering.
Boil the eggs for 8 to 9 minutes.
Transfer the eggs immediately to a bowl with cold ice water. Let it sit for 5 to 10 minutes to cool down.
After cooling down, crack the egg shells and peel them.
Separate the egg yolks from the egg whites. Add the egg yolks to a bowl and the egg whites to a chopping board.
Chop the egg whites into small pieces so it is diced.
In the bowl with the egg whites, add the Japanese mayo, salt, black pepper, sugar, mustard, and cream or milk.
Mash the yolks and mix it all together until you have the consistency of a thick sauce or paste.
Add the egg whites into the bowl and combine all together.
Prepare the bread by slicing the crusts off and spreading butter on both sides.
Add thinly sliced cucumber to one side if desired and then egg filling on top.
Seal the sandwich shut with the other piece of bread and cut in half to enjoy!