Spread one third of the seasoned rice onto a sheet of nori and scatter with one third of the sesame seeds.
Wrap your bamboo sushi mat in plastic wrap and flip the nori over, so it is now rice side down on the covered mat.
Arrange three shrimp in a line along the edge of the nori followed by one third of the cucumber.
Roll up tightly, pressing down well with the sushi mat. Unroll and remove the plastic wrap.
Place one third of the avocado slices along the top of the dragon roll. Replace the plastic wrap and roll up again with the bamboo mat, pressing firmly but gently together so the avocado curves to fit the roll.
Unroll and place on a chopping board. Repeat with the other two sheets of nori and the rest of the ingredients.
Slice each roll into 6-8 pieces of sushi.
Top each slice with a small blob of spicy mayonnaise and a little pickled ginger. Serve with soy or unagi sauce on the side to dip.