Begin by making the eel sauce. In a small saucepan, dissolve the sugar in your soy sauce and mirin. Once dissolved, remove from the heat and store in an airtight container in the fridge (should last two weeks).
Prepare your eel according to package instructions.
Fill a small bowl with water, this will be used to moisten the edge of your nori to help it stick together.
Place your first nori sheet on a cutting board and add enough sushi rice to fully cover the nori sheet. This should require ½-¾ cups of rice per roll.
Next, place 3 ounces of unagi lengthwise about an inch from the edge of your nori sheet. Follow this with 2 slices of cucumber and 2-3 slices of avocado.
To roll your sushi, carefully lift the edge of the nori with the filling and lift it, with your index fingers holding the filling in place. Then, lift and tuck the roll while continuing to roll the nori until you have reached the end of your nori sheet. This may require some practice, so be patient.
If necessary, dab the edge of your nori to help secure it into a tight roll.
Top your roll with toasted sesame seeds and a generous drizzle of eel sauce.