Wrap the bamboo mat completely in plastic wrap both lengthwise and crosswise. Place the wrapped mat on a cutting board or other flat work surface.
Prepare the crab mixture: take 8–10 imitation crab sticks from the package and cut them into small pieces. Put the pieces in a bowl, add the 2 tablespoons mayonnaise, and whisk vigorously until the crab is broken up and the mixture is evenly combined.
Prepare the avocado: cut the avocado in half, remove the pit, scoop the flesh out of the skin, and slice into ½-inch slices.
Prepare the cucumber: cut the ends off the English cucumber and slice it into sticks. You should get about 8 even sticks from one cucumber.
Place one nori sheet shiny-side down on the plastic-covered bamboo mat. With clean, slightly damp hands, take about ¼ cup of cooled sushi rice and spread it evenly over the nori, covering most of the sheet.
Flip the nori sheet over so the rice is facing down on the mat.
In the middle of the nori (on the seaweed side), place about 3–4 tablespoons of the crab mixture in a horizontal line across the sheet.
Add two cucumber sticks next to the crab line, and lay avocado slices on top of the cucumber sticks.
Starting from the bottom edge of the mat, fold the mat over the filling and roll upward, keeping firm, even pressure so the filling stays in place. Continue rolling until the nori is closed and the roll is formed.
Give the finished roll one more tight roll in the bamboo mat to compact it into an even cylinder.
Use a sharp knife to slice the roll into individual pieces. Make one straight cut per piece—do not saw back and forth.
Plate the sliced rolls and serve with soy sauce, wasabi, and pickled ginger.