Rinse the sushi rice under cold water until the water runs clear.
In a pot, combine rinsed rice and water. Bring to a boil over medium heat, then reduce to low, cover, and simmer for 20 minutes or until rice is cooked and water is absorbed.
Once cooked, remove from heat and let it sit covered for an additional 10 minutes.
In a small bowl, mix rice vinegar, sugar, and a pinch of salt until dissolved. Stir this mixture into the warm rice, allowing it to cool to room temperature.
Place a bamboo sushi mat on a clean surface. Lay one sheet of nori on the mat, shiny side down.
Wet your hands to prevent sticking, and spread about 1/4 of the sushi rice evenly over the nori, leaving a 1-inch border at the top.
Arrange a line of avocado, cucumber, carrot, bell pepper, and purple cabbage across the center of the rice.
Starting from the bottom, use the mat to gently roll the sushi away from you, pressing tightly to form a roll.
Once rolled, wet the top edge of nori with a bit of water to seal the roll.
Repeat with the remaining ingredients to make additional rolls.
Using a sharp knife, slice each roll into 6-8 pieces, wiping the knife with a damp cloth between cuts for cleaner slices.
Arrange the sushi rolls on a platter, sprinkle with sesame seeds, and serve with soy sauce and pickled ginger on the side.
Notes
Use a sharp knife for clean cuts and serve with soy sauce and pickled ginger.