Cut off top quarter of nori sheet along short end. Place nori, shiny side down, on a sushi mat covered with plastic wrap, with long end toward you. Pat 3/4 cup rice over nori with moist hands, leaving a 1-inch border on one long end of the nori.
Arrange 1/4 cup shrimp, 2 carrot strips, 2 slices avocado, 1 green onion top, and 2 cucumber strips along top third of rice-covered nori.
Lift edge of nori closest to you; fold over filling. Lift bottom edge of sushi mat; roll toward top edge, pressing firmly on sushi roll. Continue rolling to top edge; press mat to seal sushi roll. Let rest, seam side down, for 5 minutes. Repeat procedure with remaining ingredients. Slice each roll into 8 pieces with a sharp knife.
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