Place potatoes in a large pot and fill the pot with water to cover potatoes—season water with kosher salt. Bring the pot to a boil over high heat and reduce to a simmer. Cook until potatoes are fork tender, 15-20 minutes. Drain potatoes and set aside.
While the potatoes cook, prepare the ½ cucumber, grate the carrots, and hard-boil the eggs.
Combine the mayonnaise, rice vinegar, and hot mustard in a small bowl. Set aside.
Place more than half of the cooked potatoes in a large bowl and add the milk. Using a masher, roughly mash the potatoes, still leaving texture. Add the cucumbers, carrots, red onion, and green onions, and pour the dressing over the potatoes. Fold to combine.
Roughly mash the balance of the potatoes, leaving larger chunks for texture, and the chopped hardboiled eggs and fold to combine—season with salt to taste.
Serve immediately or can be made a day in advance.
Garnish with sliced green onion tops.