Cook asparagus in boiling water for 1 minute. Remove and rinse with cold water to stop the cooking. Drain and set aside.
Wash and drain rice 4 to 5 times in a medium sized pot. Cover rice with 3 cups (720ml) water. Drop in kombu and allow contents to soak for 30 minutes.
Place pot on stove over medium heat. Just as water is about to boil, remove kombu. Place the lid back on and allow water to come to a boil. Reduce heat to medium low and cook for about 20 minutes or until all water is absorbed.
Turn off stove and remove pot without opening the lid. Allow rice to sit for 10 minutes.
Combine rice vinegar, sugar, and salt in a small bowl. Stir until sugar and salt dissolve.
Transfer rice to a moistened sushi tub or large shallow bowl. Pour vinegar mixture evenly over the rice. Gently mix in the vinegar mixture with a rice paddle in a cutting motion. Do not mash. Turn the rice with the paddle and fan as you go along. This will help the rice absorb the vinegar and give it a glossy look.
Making the rolls and serving
Place a sushi mat on your work surface. Then place a sheet of nori with shiny side down onto mat, lining with the edge of the mat closest to you.
Moistened your hands and spread an even layer of vinegared brown rice with the help of a rice paddle onto nori, leaving about 1½ inches of nori at the far end. The rice should be approximately a quarter inch high. Create a ridge at the far end to prevent filling from spilling forward when rolled.
Line up a strip of cucumber, 1 asparagus, and 1½ strips of surimi fish stick in the center of the rice.
Holding down the filling with your fingers, lift the mat with your thumbs, rolling forward until the edge of the mat touches the top of the ridge.
Firmly squeeze the mat and continue rolling forward while retracting the mat backwards so that the seam of the nori is at the bottom.
Finally, place the top portion of the mat firmly over the roll, curl fingers and squeeze mat to compact the roll. Do not push too hard as to force the filling out at the ends. Repeat until all rice is used up.
Moistened knife and slice each sushi roll into 6 thick slices.
Serve with soy sauce, a small dollop of wasabi, and pickled ginger.