This is a veganized version of my Japanese potato salad recipe. Japanese-style potato salad is bit different because you roughly mash the potatoes before adding the other ingredients. You also use Japanese mayonnaise and add sliced cucumbers and carrots.
Put the potatoes, carrot, and a large pinch of fine grain sea salt in a large pot and cover them with cold water. Bring the water to a boil over medium heat and let the potatoes simmer until they are fork tender. Remove the carrot when it is starting to soften, but before it can be pierced through with a bamboo skewer. Start checking the carrot and potatoes after they’ve been simmering for about 8 minutes. When the potatoes are done, drain them well.
While the potatoes are cooking, sprinkle the cucumber slices with 1 teaspoon of fine grain sea salt, toss, and let them sit for 5 to 10 minutes, then gently squeeze them and drain out the excess cucumber water.
Slice the cooled carrots in half lengthwise, then slice each half into thin half-moons. Coarsely mash the potatoes so some small chunks of potato still remain, then let the mash cool.
After the potato mash has cooled, stir in the cucumbers, carrots, red onion, and hard-boiled eggs. Gently stir in the mayonnaise and rice wine vinegar. Add additional mayonnaise, if desired. Cover the potato salad with plastic wrap, or place it in an airtight container and let it chill in the refrigerator for at least 1 hour.