Cook the rice according to package directions. (Basically: 1/2 cup brown sushi or sweet brown rice, 1 1/4 cups water. Cover and simmer for 45 minutes, then let it sit for 10 minutes).
When the rice is done, mix in 1 tablespoon of rice vinegar.
Add the second tablespoon of vinegar to the bowl of water.
Cut the carrot and cucumber into long, thin strips.
Cut the avocado in half and remove the pit. Slice thinly and spoon out.
Place a sushi mat on the counter top or cutting board. Cover with a piece of plastic wrap. Place the nori on the wrap, shiny side down, with the perforated lines running up to down.
Scoop about half of the rice on to the nori. Wet your hands in the bowl and spread the rice evenly across 3/4 of the nori: leave the top 1/4 empty. If the rice sticks to your fingers, wet your hands again.
Sprinkle with sesame seeds.
Place carrot and cucumber strips across the center of the nori and then top with slices of avocado. Make sure the vegetables reach the edges of the nori.
Using the mat and plastic wrap, roll it up as tightly as you can and give it a little squeeze. It should stick together as the rice dampens the seaweed.
Remove the roll from the mat and plastic wrap and place on a cutting board. Cut into bite-sized pieces.
Arrange the pieces on a plate and serve with wasabi, picked ginger and soy sauce.
Covering the sushi mat with plastic wrap helps to keep it clean.
Other veggie filling options: re-hydrated shiitake mushrooms, baked tofu, blanched and cooled cabbage, bell pepper, zucchini, corn, blanched and cooled asparagus.