Line your bamboo sushi mat with plastic wrap.
Place one sheet of nori at the top edge of the mat.
Have a small bowl of water nearby. Spread 1/2 cup of rice (more if needed) over the nori, using wet fingers to press it down. Turn the nori sheet over so the rice is on the mat.
Spread 1/2 the wasabi over the nori. Layer 2 asparagus spears on the nori about 1-inch from the end closest to you. Then, place 3 of the strips of bell pepper next to the asparagus. Line up 6 shrimp next to the pepper strips. Finally, add a row of 3 avocado slices.
Starting at the end closest to you, carefully roll the nori and rice over the filling with the aid of the mat, making sure to press the mat to make the roll tight. With patience, continue to roll tightly until all the filling is rolled up. Set the roll aside, seam side down, and make the next one.
Crack the egg into a small bowl and whisk until the yolk and white are combined. Pour this into a shallow bowl large enough to hold the rolls. In another shallow bowl or plate, spread out the panko breadcrumbs.
Pour the oil into a large skillet over medium-high heat and allow it to warm up. Or, heat oil in your deep fryer.
Dip the sushi rolls in the egg and then coat them with the breadcrumbs.
Place the rolls into the hot oil and cook until golden brown and crisp. If cooking in a skillet, turn the rolls 3 or 4 times to get all surfaces golden brown. This will take about 3 to 5 minutes if the oil is hot enough.
Transfer the tempura rolls to a clean cutting board. Using a sharp serrated knife, slice the sushi rolls into 3/4-inch thick pieces.
layer the rolls on individual serving plates. Spoon the spicy mayonnaise over the pieces. Sprinkle scallions and sesame seeds over the sushi and enjoy while warm and crunchy.