Cook the sushi rice according to the instructions on the packet then set aside to cool.
Fill a large bowl with 1 cup of water and place the kanpyo in it to soak. Leave while you prepare other ingredients.
Use a potato peel or similar to peel the skin off the carrot and the cucumber. Cut the carrot and cucumber into long, thin slices with a knife.
In a mixing bowl, add the eggs and 1 tbsp sugar, salt, and 2 tbsp of water. Gently mix with a fork, but try not to overmix to avoid creating many bubbles.
Cover the base of a large frying pan with a thin layer of oil and heat over medium heat.
Pour half of the eggs into the pan and wait for it to solidify. Once it becomes fairly cooked (about 5 minutes) but still light in color, fold it in half against one side of the pan.
Add a little bit more oil to the exposed base of the pan, then pour the rest of the egg into the empty space.
Lift the already cooked portion of the egg so the runny egg fills the whole pan, then put it back down. Let cook for about another 5 minutes or until solid.
Once solid, fold the egg in half again, then transfer to a sushi mat or something similar so you can roll it up.
Set aside and let cool before cutting it into long slices.
Remove the kanpyo from the water and wrap it in a kitchen towel so you can squeeze extra water out. Cut into long, thin slices.
In a small pan, add 2 tbsp sugar, 2 tbsp of water, the rice vinegar, and the soy sauce. Place over high heat and bring to a boil.
Reduce heat to low and add the kanpyo and carrots. Let it simmer for about 15 minutes or until most of the liquid is gone and you can pierce the carrots easily with a fork.
Remove from heat and set aside to cool.
On your sushi mat, place a sheet of seaweed. On top, add about 1 ½ cup of the cooked sushi rice and press down to form an even layer on the seaweed, but leave about an inch of seaweed clear down one side.
Place some of the carrot in a stripe down the center and repeat with cucumber, kanpyo, and egg roll. Avoid piling them on top and instead arrange them nestled together.
Use your fingers to dab some water on the clear piece of seaweed. Begin rolling the other side of the seaweed. Try to press everything tightly together as you roll.
Use a very sharp knife to cut the sushi roll into roughly inch slices.
Repeat the rolling process until you have used up all your ingredients.
Serve and enjoy!