Prepare the sushi rice: Rinse the sushi rice, cook according to the package directions, then gently mix in rice vinegar, sugar, and salt while the rice is warm. Cool to room temperature.
Warm the unagi: Heat the cooked unagi kabayaki according to the package directions. Let it cool slightly, then cut it into thin strips.
Slice the avocado: Cut the avocado in half, remove the pit, peel it, and slice the flesh into thin, even strips.
Prepare the sushi mat: Cover the bamboo mat with plastic wrap if making an inside-out eel avocado sushi roll.
Spread the rice: Place half a sheet of nori on the mat and spread a thin, even layer of seasoned sushi rice across it.
Add sesame seeds: Sprinkle toasted sesame seeds lightly over the rice if desired.
Add the filling: Place the cooked unagi and avocado in a neat line across the center of the nori.
Roll the sushi: Lift the edge of the mat and fold the rice and nori over the filling. Roll forward with firm but gentle pressure.
Shape the roll: Give the finished eel avocado roll a gentle squeeze through the mat to create an even cylinder.
Slice the sushi: Lightly wet a sharp knife and cut the roll into 6–8 pieces. Wipe the blade between cuts for cleaner slices.
Finish the roll: Drizzle a small amount of unagi sauce over the sliced sushi and add sesame seeds if desired.
Serve: Serve the unagi avocado roll immediately with soy sauce, pickled ginger, and wasabi.