Eggs - Boil water in a medium pot on medium heat. When the water starts boiling, turn the heat to medium-low and add eggs. Boil for 10 minutes.
Eggs - Prepare an ice water bowl. After 10 minutes, turn the heat off and move the eggs to the bowl of ice water. Leave them for a few minutes.
Eggs - Remove the shells. It is easy to remove the shells after tapping the eggs on a kitchen counter a few times and breaking the shells into small pieces. Pat with a paper towel to remove water.
Cucumber - Slice cucumber thinly.
Cucumber - In a bowl, add the sliced cucumber and add ¼ of salt. Mix everything.*2
Other vegetables - Cut carrot and onion thinly then cover with cling wrap lightly. Microwave them for 1 minute to make them softer.
Potatoes - Peel the potatoes and then cut them into about 1 inch cubes.
Potatoes - Prepare a medium bowl and add enough water to cover the potatoes. Then, add potatoes for at least 10 minutes.*1
Potatoes - While soaking the potatoes, boil a pot of water. After 10 minutes of soaking the potatoes, remove the water from the bowl. With the water boiling, add the potatoes to the pot for 15 minutes.
Potatoes - After 15 minutes, try to stick the potatoes with a fork (or toothpick), and if the fork goes in smoothly, the potatoes are ready!
In a medium or large bowl, add the warm potatoes and mash them. It is OK not to mash until smooth.
Add eggs, salt, and rice vinegar in the bowl and mash them.
Add the onion, carrot, cucumber in the bowl. Mix them by folding them in so you don't smash them.
Add Japanese mayonnaise and mix thoroughly.
Lastly, add black paper to finish!