Shred the carrot using a julienne peeler or coarsely grate with a box grater. Thinly slice the radish. Peel the cucumber into ribbons with a vegetable peeler.
Step 2 — Pickle vegetables
Put 1 tbs white vinegar, 1 tsp sugar and a pinch of salt in a large bowl and stir to dissolve the sugar. Add the carrot, radish and cucumber and toss to combine. Set aside to pickle, tossing occasionally, until needed.
Step 3 — Cook rice
Rinse the rice until the water runs clear. Put in a small saucepan with 250ml (1 cup) water, cover and bring to a simmer over medium heat. Reduce the heat to low and cook for 12 mins or until tender and the water is absorbed. Turn off the heat and stand, covered, for at least 5 mins.
Step 4 — Crumb chicken
Meanwhile, combine the aioli and 2 tsp soy sauce in a bowl. Put the breadcrumbs in a shallow dish. Put the chicken flat on a board, put your hand on top and halve horizontally into fillets. Put 1 tbs aioli mixture in a separate bowl and brush or spoon all over the chicken, reserving the remaining aioli mixture. Press the chicken into the breadcrumbs to coat both sides.
Step 5 — Cook chicken
Heat 1cm oil in a medium frypan over medium heat. Cook the chicken for 2-3 mins each side until golden and cooked through. Drain on a wire rack set over paper towel (this will keep the crumb crisp). Season with salt.
Step 6 — Get ready to serve
Drain the vegetables, reserving the pickling liquid. Whisk the wasabi (wasabi is spicy, so reduce the amount or omit it, if you prefer) and 1 tsp pickling liquid* into the reserved aioli mixture. Fluff the rice with a fork. Divide the rice, chicken and pickled vegetables among plates. Scatter with the sesame seeds and serve with the wasabi aioli.