Cut Takuan pickled yellow radish into thin matchsticks. Set aside.
Make seasoned rice. Put steamed rice in a bowl, and season with salt and sesame oil. Cover and set aside.
Make cooked burdock. Cut the burdock root into thin strips. In a medium pot or frying pan, heat sesame oil, add burdock, and stir-fry for 2-3 minutes. Add the rest of the ingredients, cover, and cook at medium low heat until the liquid is almost gone. Set aside.
Make cooked carrot. Peel and cut carrot into thin matchsticks. In a frying pan, heat oil at medium heat, add carrot, and cook until carrot is tender. Add salt and sesame seeds, and stir. Set aside.
Make cooked spinach. In a boiling water, cook spinach for 30 seconds or so. Strain and let cool. Once it's cooled, squeeze out water very well. Cut into 1" (2.5 cm) length. In a medium bowl, add the rest of the ingredients, and mix with the cut spinach.
Make Tamagoyaki rolled egg. Beat eggs and seasonings together in a bowl. Heat a rectangular pan over medium-high heat and add oil. Pour a thin layer of egg mixture in the pan, tilting to cover the bottom. After the thin egg has set a little, gently roll into a log. Repeat adding egg to the pan and rolling back and forth until all the egg mixture is used. Refer to our Tamagoyaki recipe for guidance. When cooled, cut Tamagoyaki into thin, long strips, about 8" long and 1/2" thick (20 cm x 1 cm).
Assemble. Place a sushi mat flat on your work surface with the bamboo slats running left to right, so you can roll the mat away from you. Place a piece of seaweed on the sushi mat with one of the seaweed's sides close to the front edge of the mat (the edge near you). Spread about 1 cup of the seasoned rice on the seaweed, leaving a 1" (2.5 cm) space at the far edge. Horizontally place Takuan, burdock, carrot, spinach, and Tamagoyaki next to each other in the middle of the rice. Holding the filling down, roll from the front end of the mat toward the other end. Tighten the roll like a jelly roll, pulling the mat to compress. Remove the roll from the mat. Repeat to make another roll.
Brush the surface of the rolls with sesame oil, and sprinkle with sesame seeds. Cut the rolls into 3/4-inch (2 cm) thick slices.