Cut the cucumber in half lengthwise and trim the ends. Scrape the seeds from each half using a spoon. Cut each half lengthwise into three long strips, for a total of six strips. Put the cucumber strips on a platter, sprinkle with salt, and toss to coat. Set aside.

In a small bowl, whisk together the eggs and a sprinkle of salt. Lightly oil a 10-inch nonstick pan and cook over medium heat until warmed but not scorching hot. Cook until half of the egg has set on top, stirring to coat the entire bottom of the pan. Carefully remove the egg round with a rubber spatula and place it on a work surface to cool. If necessary, lightly re-grease the pan and continue with the remaining egg. Clean the skillet.

Cut the egg into thin slices lengthwise. Set aside.

Cucumber should be rinsed under cold running water and patted dry with towels, squeezing lightly to remove extra moisture. Wipe down the same skillet with extra vegetable oil and heat over medium-high heat until shimmering. Cook until the cucumber is barely softened, about 1 minute. Place aside to cool.

Cook, swirling and tossing gently, until the ham strips are cooked through, 1 to 2 minutes. Place aside to cool.

Cook, swirling and tossing constantly, until the carrot is crisp-tender, 1 to 2 minutes; if necessary, add a little more oil to the pan at any stage. Place aside to cool.

Reduce the heat to medium. Cook, tossing and turning, until the fishcake has softened, 1 to 2 minutes, with the oligosaccharide syrup and soy sauce in the pan. Place aside to cool.

Combine rice, sesame oil, crumbled sesame seeds, and 1 teaspoon salt in a large mixing basin (see note). Make 6 equal amounts of rice.

On a work area, place a kimbap/sushi rolling mat. Place a sheet of nori on top. Using plastic gloves, lay a quantity of rice on the seaweed sheet and distribute it evenly all over, leaving approximately one or two inches of the top edge exposed. Place a few rice grains around the top border of the dish (they will help it stick together later).

Put 1 cucumber stick, some egg strips, 1 ham strip, one-sixth of the carrot, some of the fishcake strips, 1 danmuji strip, 1 burdock strip, and 1 crab stick strip across the middle of the rice.

Roll the bamboo mat up and away from you, wrapping the seaweed sheet and rice around the fillings; keep the filling in place with the fingers of both hands as you roll with your thumbs. Use the uncovered flap of the seaweed sheet to secure the roll. Once the roll is sealed, gently squeeze it, pushing softly on the top and sides, to slightly compress it and close the edge; if it doesn't seal properly, moisten the seaweed at the seam to help it adhere.

Proceed with the rest of the seaweed sheets, rice, and fillings.

Brush sesame oil over rolls and lightly massage a sharp knife blade with it. Place two rolls next to each other, then cut them into 1/2 inch thick pieces (it's simpler to cut through two rolls at once than one at a time).

Serve.