Slice the sashimi-grade ahi tuna into long, thin strips approximately half an inch thick.
In a small bowl, gently fold the rice vinegar into the cooked sushi rice until the grains are evenly coated and slightly sticky.
Place one nori sheet on a bamboo rolling mat and spread half of the seasoned rice evenly over the bottom two-thirds of the sheet.
Layer half of the tuna strips, julienned cucumber, avocado slices, and finely chopped scallions across the center of the rice.
Carefully roll the nori from the bottom up, using the mat to maintain firm pressure, and seal the top edge with a small drop of water.
Repeat the assembly for the second roll, then slice each roll into six even pieces using a very sharp, moistened knife.
Whisk together the tamari, toasted sesame oil, fresh lime juice, and grated ginger in a small ramekin to create the dipping sauce.
Garnish the sliced rolls with black sesame seeds and serve immediately with the zesty ginger-lime sauce on the side.