Make the kabayaki sauce by adding the soy sauce, sake and sugar to a small pan and boiling until most of the liquid has evaporated and the mixture is thick and syrupy.
You're going to roll your Caterpillar Roll with the rice on the outside, so you need to cover your makisu (bamboo mat) with plastic wrap to keep the rice from sticking. Prepare a small bowl of water to dip your fingers in to keep the rice from sticking to them.
Carefully fold your 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.
Lay one piece of nori towards the bottom of the mat. Lightly wet your fingers with water, then add a small amount of rice onto the nori.
Make sure your fingers are moist, then use your fingertips to gently spread the rice out to the edges of the nori 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 cucumbers along the bottom edge of the nori, then top with the unagi.
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 avocado slices over the roll making sure each thin slice of avocado overlaps the next.
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.