Rinse the sushi rice under cold water until the water runs clear. Combine the rice and water in a rice cooker or pot, and cook according to the rice cooker instructions or bring to a boil, then reduce heat and simmer for 20 minutes. Remove from heat and let it sit covered for 10 minutes.

In a small saucepan, combine rice vinegar, sugar, and salt. Heat over low heat until the sugar dissolves. Once the rice is done, transfer it to a large bowl and gently fold in the vinegar mixture. Allow the rice to cool to room temperature.

While the rice cools, prepare the tempura shrimp. In a bowl, mix the tempura batter mix with cold water until just combined. Heat oil in a deep pan to 350°F. Dip each shrimp in the batter and fry until golden brown, about 2-3 minutes. Drain on paper towels.

Lay a bamboo sushi mat on a flat surface. Place a sheet of nori, shiny side down, on the mat. Wet your hands to prevent sticking, then spread about 1/2 cup of sushi rice evenly over the nori, leaving a 1-inch border at the top.

Arrange a few strips of mango, avocado, and 2-3 tempura shrimp horizontally across the rice. Using the bamboo mat, carefully roll the sushi away from you, pressing gently but firmly to keep the roll tight. Seal the edge of the nori with a little water.

Repeat the rolling process with the remaining ingredients. Once all rolls are made, use a sharp knife to slice each roll into 6-8 pieces. Wipe the knife with a damp cloth between cuts for cleaner slices.

Arrange the sushi on a serving platter. Sprinkle with sesame seeds and chopped green onions. Serve with soy sauce, wasabi, and pickled ginger.