Rinse 1 cup short-grain sushi rice under cold water until the water runs clear.
Cook the rice with 1 1/4 cups water according to your rice cooker’s instructions or on the stovetop, then let it sit covered for 10 minutes.
Meanwhile, whisk together 2 tablespoons rice vinegar, 1 tablespoon granulated sugar, and 1/2 teaspoon salt until dissolved.
Gently transfer the warm rice to a large non-metallic bowl and fold in the vinegar mixture with a rice paddle or wooden spoon without mashing the grains.
Let the seasoned rice cool to room temperature.
Preheat your broiler or oven to 375°F and warm 4 ounces pre-cooked unagi according to package directions until heated through and slightly crispy.
Slice the warmed unagi into thin strips.
Slice 1 ripe avocado thinly.
Cut 1/2 English cucumber into thin sticks.
Place 1 sheet nori shiny side down on a bamboo rolling mat.
Moisten your hands and spread about 3/4 cup of cooled sushi rice evenly over the nori, leaving a 1/2-inch border at the top.
Sprinkle a few sesame seeds over the rice, if desired.
Flip the nori sheet so the rice faces down and the seaweed faces up.
Arrange cucumber sticks and unagi strips across the center of the nori.
Using the bamboo mat, roll the nori and fillings away from you, applying gentle, even pressure to form a firm cylinder.
Lay the avocado slices over the top of the roll, overlapping them to create a “dragon scale” effect.
Optionally, cover the avocado with plastic wrap and gently press with the mat to help it adhere.
If you have extra unagi, lay additional strips atop the avocado.
With a very sharp, slightly moistened knife, slice the roll into 8 equal pieces, wiping the knife clean between cuts.
Serve immediately with soy sauce, pickled ginger, and wasabi.