Bring 4 cups water to a boil in a medium saucepan. Add the quinoa, reduce the heat to a simmer and cook until the pan is dry and the quinoa is tender and sticky like sushi rice, about 30 minutes. Remove from the heat, drain and mix with 1 tablespoon of the vinegar, 1 of the teaspoon shoyu and 2 teaspoons of the brown rice syrup. Set aside until ready to assemble.
Combine the gochujang, 2 tablespoons of the shoyu, the remaining 1 tablespoon vinegar and the remaining 2 teaspoons brown rice syrup in a small bowl. Remove 1 tablespoon of the marinade for the dipping sauce and set aside.
Lay the tofu flat. Cut off one-third of the tofu block and reserve for another use. Turn the remaining block on its side, cut it into 3 slices and place in a small glass baking dish. Spoon the marinade all over the tofu. Cover and refrigerate for 30 minutes (or overnight).
Cut each tofu slice into 4 planks, for a total of 12. Cut the avocado into 1/4-inch strips and coat with the lime juice to prevent it from browning. Cut the cucumber into thirds, then standing each vertically, slice two 1/4-inch planks off the sides and discard the middle with the seeds. Cut each plank into 4 thin strips.
Place a sheet of nori on a dry surface and evenly spread 1/2 cup quinoa over it, leaving a 1-inch border at the top. Sprinkle 1 teaspoon of the sesame seeds over the quinoa. Lay 2 pieces of tofu, 3 avocado slices and 4 cucumber strips in a straight horizontal line at the end closest to you. With dry hands, roll the nori sheet away from you, tightening as you go. As you reach the border at the top, dab your fingertip in water and moisten the top edge of nori to seal the roll. Using a sharp knife, cut into 6 equal pieces. Repeat to make 5 more rolls.
Make a dipping sauce by combining the reserved 1 tablespoon marinade with the remaining 2 tablespoons shoyu. Serve the sushi rolls with the dipping sauce.
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