At Korean barbecue restaurants, small dishes of this style of mashed potato salad are served with the grilled meat. Cool and creamy with mayonnaise and boiled eggs, it’s the perfect accompaniment to the smokiness of charred meat. Pops of crunch from quick-pickled cucumber add a refreshing note. Starchy Russet or other baking potatoes smash the most easily here; waxy ones, such as red, white or Yukons, will give you a chunkier salad. While this is ideal with homemade mayonnaise, it’s also tasty with store-bought varieties, especially the Japanese Kewpie brand.
Place the cucumber in a small bowl and sprinkle with ½ teaspoon sugar and 1 teaspoon salt. Toss to evenly coat and let stand until ready to use.
Combine the potatoes and 1 tablespoon salt in a large saucepan. Add enough cold water to cover by 2 inches. Bring to a boil over high heat, then reduce the heat to maintain a light boil. Using a spoon, lower the eggs into the boiling water. Boil the potatoes and eggs together for 7 minutes, then add the carrot. Continue boiling for 5 minutes, then transfer the eggs to a small bowl with a slotted spoon and add enough ice to cover. Boil the potatoes and carrots until the potatoes are very tender, about 1 minute longer. Drain well, then return to the dry saucepan.
Set the saucepan over the lowest possible heat to dry out the potatoes. When they’re speckled and starting to stick to the pan, about 1 minute, transfer to a large bowl. Sprinkle the vinegar and remaining ½ teaspoon of sugar evenly over the potatoes and toss to coat. Mash about half the potatoes and break up the remaining into smaller chunks. Transfer to the refrigerator to cool.
Peel the cooled eggs and coarsely chop, then add to the potatoes. Grab the cucumber slices by the handful to squeeze out all the liquid and add to the potatoes. Add the mayonnaise and gently fold until well-mixed, then season to taste with salt and pepper. Serve immediately or refrigerate to serve cold.
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