Cook the sushi rice according to the package or to the rice cooker manufacturer's instructions

Heat the rice vinegar, sugar, and salt either in the microwave or over the stove and stir until the sugar has dissolved.

Pour the mixture over the sushi rice, fanning constantly and gently folding/stirring the rice until the mixture is absorbed in the rice. Be careful not to break any of the grains. The rice should be tacky and fragrant. You can either add the toasted sesame seeds here, or sprinkle it onto the rolls later.

To make the sushi rolls, gently spread some of the sushi rice over the nori, leaving space at one end to make the seal. It's best to wet your hands with water to handle the rice so you don't get monster rice hands. Do not squish or pack the rice, you just want to make a light layer.

Arrange the cucumber and avocado in the bottom third of the square, then start to roll. Use the sushi mat to roll and shape the ingredients into a roll.

Continue rolling and shaping with the mat until you get a nicely shaped roll. Use a bit of water to seal the end. Cut into sushi pieces with a sharp knife (either into 6 or 8 pieces, whichever you prefer). I cut usually by cutting the whole roll in half, then placing the two halves side by side, and then cutting those halves into four pieces.

To make the hand roll, cut the nori squares in half to get two rectangles.

Spread a little bit of rice (about 2-3 tbsp) to one end of the rice in sort of an oval-ish clump.

Add a piece of avocado and cucumber and roll into a cone, using a bit of water to seal the end.

Serve sushi with soy sauce and wasabi (and pickled ginger if that's your thing). Best served immediately, because even with the lemon juice the avocado will brown. It'll taste fine but it won't look that pleasing.