Cook the rice according to the package directions.
Meanwhile, put the sugar, 1 teaspoon salt and 4 tablespoons of the vinegar in a medium microwave-safe bowl and microwave until hot, about 30 second. Stir to dissolve the sugar and salt. Let cool completely.
Gently fold the vinegar mixture into the rice (reserve the bowl for the tuna). Let cool to room temperature, uncovered.
In the reserved bowl, whisk together the mayonnaise, a pinch of salt and pepper and the remaining 2 tablespoons vinegar. Stir in the tuna and scallions.
Put a 20-by-15-inch piece of parchment on a clean work surface. Place 4 sheets of the nori shiny-side down on the parchment to create a 15-inch square, overlapping them by about 1/4 inch. Lightly brush the overlapping edges of the nori with water and press them together to help seal them.
Dampen your hands with a bit of cold water and spread half of the rice in an even layer all over the nori, leaving a 2-inch border around the edges. Sprinkle half of the furikake evenly over the rice. Lay half of the carrot, cucumber, pepper and avocado on top of the furikake horizontally along the bottom third of the rice, not going onto the nori. Spoon half the tuna on top of the vegetables.
Wrap like a burrito: fold the nori edge closest to you over the filling. Fold the 2 sides inward to seal the ends and roll the whole thing over once more to completely seal; use a little water if needed to help close the seal. Wrap the burrito up tightly in a large piece of parchment or wax paper. Repeat this process to make a second burrito with the remaining ingredients. Cut the burritos in half on the bias and serve.
Furikake rice seasoning is a blend of dry ingredients that may include sesame seeds, nori flakes, bonito or fish flakes, dried wasabi, salt, sugar and other spices. It can be found in specialty grocery stores or online. If unavailable, a mixture of black and white sesame seeds is a good substitute.