Rinse the sushi rice under cold water until the water runs clear. Combine the rice and water in a rice cooker and cook according to the manufacturer's instructions. Once cooked, let it sit covered for 10 minutes.

In a small saucepan, combine rice vinegar, sugar, and salt over low heat. Stir until sugar and salt dissolve. Pour the mixture over the cooked rice and gently fold to combine. Allow the rice to cool to room temperature.

In a mixing bowl, combine the tempura batter mix with cold water until smooth. Heat oil in a deep pan to 350°F. Dip the shrimp in the batter and carefully fry until golden brown, about 2-3 minutes. Remove and drain on paper towels.

Place a bamboo sushi mat on a flat surface. Lay one 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 pieces of tempura shrimp horizontally across the rice.

Using the bamboo mat, carefully roll the sushi away from you, pressing gently to form a tight roll. Seal the edge of the nori with a little water.

Repeat the process with the remaining ingredients. Once all rolls are made, slice each roll into 6-8 pieces using a sharp knife dipped in water.

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