Fill a saucepan large enough to cover the potatoes with water and add 1 tablespoon of the salt. Bring to a boil over high heat. Reduce to medium and cook the potatoes for about 10 minutes, or until a skewer goes through them easily.
Drain and coarsely mash the potatoes so small chunks are still visible. Set aside and allow to come to room temperature.
Add the cucumber, carrot, onion, and 1 teaspoon of the salt to a bowl. Mix well with your hands, coating everything with salt. Allow to cure for 5 minutes. Add the water and stir to rinse off the salt. Squeeze the cured vegetables tightly to expel the liquid.
Add the vegetables and potatoes to a large bowl and mix together. Add the vinegar and mix to combine. Add the mayonnaise, pepper, and the remaining 1/2 teaspoon salt. Mix until smooth and serve.