Rinse 240ml short-grain sushi rice under cold water until the water runs clear.
Combine the rice and 300ml water in a saucepan. Bring to the boil, then reduce the heat to low. Cover and simmer for 12-15 minutes, or until the water is absorbed. Remove from the heat and leave, covered, for 10 minutes.
Meanwhile, whisk together 30ml rice vinegar, 15ml granulated sugar and 2.5ml salt in a small bowl until the sugar and salt dissolve.
Transfer the warm rice to a large non-metallic bowl. Gently fold in the vinegar mixture with a rice paddle or wooden spoon, taking care not to mash the grains. Leave to cool to room temperature.
Preheat the grill in your oven to high, or set the oven to 190°C (fan 170°C/Gas mark 5) on the grill setting.
Place about 110g pre-cooked unagi (broiled eel) on a baking tray and grill for 3-5 minutes, or until heated through and slightly crisp. Remove from the oven and slice into thin strips.
Thinly slice 1 ripe avocado and cut half a cucumber into thin sticks.
Place a sheet of nori, shiny side down, on a bamboo sushi mat. Wet your hands and spread about 180ml of the cooled sushi rice evenly over the nori, leaving a 1cm border at the top. Sprinkle with sesame seeds, if using.
Flip the nori so the rice side is down and the seaweed side is up. Arrange the cucumber sticks and unagi strips along the centre.
Using the bamboo mat, roll the nori and filling 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 resemble dragon scales. Cover with cling film and gently press with the mat to help the avocado adhere.
If you have any extra unagi, place the remaining strips on top of the avocado.
Using a very sharp knife dipped in water, slice the roll into 8 equal pieces, wiping the blade clean between cuts.
Serve immediately with soy sauce, pickled ginger and wasabi.