Make the teriyaki tofu first, cut the tofu into 1/4 inch slices;
Pat the slices dry, coat the tofu in arrowroot powder or cornstarch, shake off any execess powder;
Heat a drizzle of avocado oil in a nonstick pan over medium-high heat, pan fry the tofu slices for 3-4 minutes or until golden brown, flip, and fry the other side till golden brown;
Meanwhile, mix together all the teriyaki tofu sauce ingredients, lower the heat to low, slowly pour in the sauce and move the tofu slices around in the sauce as it thickens to coat all the sides evenly;
Remove from the heat and set aside;
Season the cooked sushi rice with sushi vinegar;
Add a few drops of water to the vegan cheese in a bowl, melt in the microwave by heating for 30 seconds, check to see if all the shreds are melted, add 10 seconds increment if needed;
Lay down a piece of nori on your work surface, add 3/4 up of the sushi rice, press down in the center to form a square;
Layer on teriyaki tofu, melted vegan cheese, carrots, pickled cabbage, and avocado;
Press the rest of the 1/3 cup sushi rice in your hand to form a disc, add on top of the fillings, pick up all four corners of the nori sheet and wrap upward to cover as much as you can;
Then add the second piece of nori on top, cover any gaps on the side that the bottom nori didn't cover, shape the onigiri firmly with your hands;
Dip the rice paper in water for 2-3 seconds, sprinkle some sesame seeds on top, transfer the onigiri to the center of the rice paper, then wrap and seal the whole rice ball;
Mix together the dip ingredients for serving;
Slice the sushi burger in half, eat with the dipping sauce, enjoy!