Cook the rice according to package instructions. Season with salt and pepper, fluff with a fork, and set aside to cool.

Make Korean Potato Salad Bring a pot of water to boil. Add potatoes, boil for 2 minutes, then add carrot chunks and eggs, cooking an additional 8 minutes until tender. While vegetables boil, sprinkle salt over cucumber and onion; let sit for 10 minutes, then rinse and squeeze out excess water.

Drain potatoes and carrots well; cool slightly. Peel eggs, separating whites from yolks. Dice egg whites and finely mash yolks separately. In a bowl, gently combine mayonnaise, onion, cucumber, carrots, diced egg whites, and apple chunks. Season with salt and pepper, garnish with mashed yolks, and refrigerate until serving.

Melt butter in a nonstick skillet over medium-low heat. Crack eggs into skillet. When whites are mostly set, carefully flip eggs, lightly dust both sides with flour, and drizzle soy sauce mixed with maple syrup over eggs; cook 1 minute more. Remove eggs from skillet and cut them in half.

Lay a sheet of nori on a bamboo sushi mat (or towel lined with plastic wrap). Evenly spread cooled rice on nori, leaving a 1-inch border uncovered along edges.

Arrange cheese slices, avocado slices, mayonnaise, and prepared eggs onto the rice. Firmly roll sushi using the mat, pressing gently to seal ingredients tightly. Slice sushi roll into 6-8 even pieces, wiping blade clean between slices.