Place freshly made rice in a large bowl. Gently mix salt and sesame oil over top with a rice scoop. Let it cool down enough so it's no longer steaming. Cover and set aside.
Tuna Mayo
Drain the tuna from the water or oil with a sieve. Press the tuna with a spoon to squeeze the excess water or oil out.
In a small bowl, combine drained tuna with mayonnaise until well incorporated. Set aside.
Egg Strips
In a medium bowl, beat eggs with salt.
Heat a little bit of oil on a big non-stick pan with low-medium heat. Pour the egg mixture into the pan with low heat. Spread it into a large circle so it fills the pan. When the bottom of the egg is cooked, flip it over with a spatula. Remove from the heat and let it cook slowly on the hot pan for about 3-4 minutes.
Transfer the cooked egg to a plate. When it is cooled enough to handle, cut it into 1/2-inch long strips.
Vegetables
Heat the pan and add a few drops of vegetable oil. Add carrot and 1/4 tsp salt, saute until the carrot is cooked. Stir in 1/2 tsp sesame oil until well combined. Set aside.
Bring a large pot of water to a boil over high heat and prepare an ice bath. Blanch the spinach, place immediately in the ice bath, then squeeze out water. Season with 1/2 tsp sesame oil and 1/4 tsp salt. Set aside.
Assemble
Place a sheet of nori on a bamboo mat with the shiny side down. Spread evenly about 3/4 cup cooked rice over the top, leaving about 1 inch uncovered on top of the nori. Place tuna mayo, 2-3 egg strips, carrot, spinach, and a strip of danmuji in the center of the rice.
Use both hands to roll the mat over the fillings. Remove the roll from the mat. Repeat 4 more times with the remaining ingredients.
Brush sesame oil on the finished rolls. Cut each roll into 1/4-inch bite-size pieces with a sharp knife, occasionally wiping it with a wet paper towel or cloth to clean the starch off and to ease cutting. Put it on a plate and serve immediately or pack it in a lunchbox.