In a medium bowl, combine the cooked sushi rice, rice vinegar, sugar, and salt. Mix well and let it cool slightly.
Lay a sheet of nori on a clean surface or bamboo sushi mat. Wet your hands to prevent sticking, then spread about ½ cup of the sushi rice evenly over the nori, leaving a 1-inch border at the top.
Arrange a few slices of avocado, cucumber, and shrimp or crab meat in a line across the center of the rice.
Carefully roll the nori away from you, tucking it tightly around the filling. Use the border to seal the roll. Repeat with the remaining nori sheets and filling.
Once all rolls are made, slice each roll into 6 pieces.
Set up a breading station: dip each sushi piece in the beaten egg, then coat with panko breadcrumbs.
In a large skillet, heat about ½ inch of vegetable oil over medium heat. Once hot, carefully add the sushi pieces in batches, frying until golden brown on all sides, about 2-3 minutes per side.
Remove the fried sushi from the skillet and place on a paper towel-lined plate to drain excess oil.
Serve warm with soy sauce, wasabi, and pickled ginger if desired.