Make the kabayaki sauce by adding the ¼ cup soy sauce, ¼ cup sake, and ¼ cup evaporated cane sugar to a small saucepan and bringing it to a boil. Cook it until most of the liquid has evaporated, and the mixture is thick and syrupy.
¼ cup soy sauce, ¼ cup sake, ¼ cup evaporated cane sugar
You're going to roll your Caterpillar Roll with the rice on the outside, so you need to cover your makisu (bamboo sushi mat) with plastic wrap to keep the rice from sticking. Prepare a small bowl of cold water to dip your fingers in to keep the rice from sticking to them.
Carefully fold your sheet from 1 pack unseasoned nori in half. If the nori is fresh, it should easily split in half along the fold and make two 3.75 inch x 8 inch pieces. If it's not splitting easily, use scissors to cut the nori in half.
1 pack unseasoned nori
Lay one piece of nori towards the bottom of the mat. Lightly wet your fingers with water, then add a small amount of 1 batch prepared sushi rice onto the nori.
1 batch prepared sushi rice
Make sure your fingers are moist, then use your fingertips to gently spread the rice out to the edges of the nori sheet in a thin even layer. Don't use too much pressure, or you'll end up mashing the grains of rice together.
Flip the rice and nori over so that the rice is facing down and the nori is facing up. Place some of the 1 hot house cucumbers along the bottom edge of the nori, then top with some of the 2 packs unagi.
2 packs unagi, 1 hot house cucumbers
Tuck your thumbs under the bamboo mat, then use the rest of your finger to hold the filling in place. Roll the mat up and over the filling.
When the mat has come all the way around the roll, you'll need to keep rolling with one hand, while using the other to lift the mat out of the way so you don't roll it into your sushi.
Once the Caterpillar Roll is fully rolled, give the whole thing a hug with your fingers. This will compress the rice, which will help keep it from falling apart when you slice it.
Splay the slices of 2 avocados over the roll, making sure each thin slice of avocado overlaps the next.
2 avocados
Wrap the roll with the bamboo mat again, and give it one more hug.
Transfer the finished Caterpillar Roll to a cutting board and use a long, sharp knife (preferably a sushi knife) to slice the roll into 8 pieces. Start slicing the roll by putting the back edge of the knife on the roll and pulling the knife towards you, using the weight of the knife to slice through the roll. If you press down, you will smash the roll.
Plate the roll and drizzle with the kabayaki sauce. You can garnish with two sprouts (antennas) and some sesame seeds if you want, but they're not necessary. I also like dusting a little sansho powder on top.