Peel the potatoes and cut into small pieces. Store them under cold water until it's time to use.
Bring the egg to a boil in salted water and cook until it is hard-boiled.
Boil the potato pieces in salted water until they become soft enough to mash.
Cut the end off of the Japanese cucumber and rub each surface together for about 10 seconds, and then rinse the cucumber under cold water.
Chop the cucumber into small pieces. Sprinkle a little salt, and leave them for 5-10 minutes to remove excess moisture.
Once the egg is hard-boiled, let it rest in cold water for a few minutes. Then peel, and chop it into small pieces.
Squeeze the cucumber pieces inside a dry paper towel to remove the excess moisture.
Cut 2 slices of cooked ham into small pieces. Make sure you have roughly equal portions of the cucumber, egg, and ham.
Check if the potato pieces are soft by testing the insides with a thin stick. When they are soft enough to be mashable, drain the boiled water, making sure there is none left in the pot.
Mash the potatoes while they are still warm. Then add 1/2 tsp each of salt and freshly ground black pepper, 1/3 tsp of Japanese rice vinegar, and 3 tbsp of Kewpie mayonnaise. Mix all together well.
Add the egg, ham, and cucumber to the seasoned mashed potato, and mix together thoroughly. It is ready! We hope you enjoy your "TOKYO FLAVORS, NEW YORK STYLE" meal! Itadakimasu!