Place potatoes in a large pot and add enough cold water to cover by 1 inch. Season generously with salt and bring to a boil over high heat. Reduce to a simmer and cook until potatoes are easily pierced with a fork, 10 to 15 minutes. Drain potatoes and transfer to a large bowl. Let cool.
Meanwhile, place cucumbers in a bowl and sprinkle ½ teaspoon salt. Mix well, then let stand for 10 minutes. Drain any accumulated water, then gently squeeze cucumbers with paper towels to blot up any excess liquid.
In a small bowl, whisk together rice vinegar, mayonnaise, and hot mustard until well combined.
Mash potatoes with a masher or a large fork, allowing some small lumps to remain. Add cucumbers, carrots, onions, eggs, scallions, and mayonnaise mixture. Mix well and season with salt to taste. Serve right away. The salad can be refrigerated up to 1 day.
