Place the peeled and cubed potatoes in a large pot and cover with cold water. Add 1 teaspoon salt and bring to a boil. Reduce to a simmer and cook until the potatoes are fork-tender, about 10-12 minutes.
Drain the potatoes in a colander. For best texture, prepare an ice bath and cool the potatoes for 2-3 minutes, then drain thoroughly.
Transfer the potatoes to a large mixing bowl. Using a potato masher or fork, gently mash about half the potatoes, leaving plenty of chunks for texture.
In a separate bowl, whisk together mayonnaise, gochujang, rice vinegar, and sugar until smooth. Season with salt and pepper to taste.
Pour about three-quarters of the dressing over the slightly warm potatoes and fold gently to combine.
Add the sliced cucumber, shredded carrot, and most of the green onions (reserve some for garnish). Fold together gently until evenly distributed.
Add the remaining dressing if needed. Taste and adjust seasonings as desired.
Transfer to a serving dish and sprinkle with toasted sesame seeds and reserved green onions. For best flavor, let the salad rest at room temperature for 15-20 minutes before serving.
