Team creamy avocado with your favorite crunchy vegetables -- we used julienned cucumbers and carrots -- in these DIY sushi rolls. For our food editor's tips on how to roll like a pro, see How to Make Sushi at Home.
Rinse rice in several changes of water, jostling rice with fingertips, until water runs clear. Set in a sieve over a bowl to drain while you prepare rest of ingredients (at least 20 minutes).
Using a mandoline or a vegetable peeler, thinly slice ginger along the grain into paper-thin slices (to yield 1/2 cup). Bring a small pot of water to a boil, add ginger, and blanch 5 seconds; drain. Return pot to stove and add 3/4 cup vinegar, 2 tablespoons sugar and 2 tablespoons water. Season lightly with salt. Bring to a simmer, stirring to dissolve sugar. Reserve half the vinegar mixture (this will be used to season your rice); add blanched ginger to remaining vinegar. Cool completely.
Place rice and 2 1/2 cups water in a small pot. Bring to a boil, then reduce heat to lowest setting and cover. Cook until rice is tender and all the water has been absorbed, about 12 minutes (alternatively, cook rice in a rice cooker according to manufacturer's instructions). Remove from heat. Let sit with lid on, 10 minutes more. Transfer rice to a large bowl; cool 5 minutes. Fold in one-quarter of the reserved vinegar mixture with a wide spatula or plastic bench scraper, fanning the rice while you fold in vinegar mixture to cool it down quickly. Cover with a clean kitchen towel. Reserve remaining vinegar mixture for assembling.
While rice cooks, toss cucumbers and carrots with remaining 1/4 cup vinegar, 1 teaspoon sugar, 1 teaspoon salt, and sesame oil.
To assemble, place 1 sheet of nori, shiny-side down, on a bamboo rolling mat with the long side closest to you. Dampen hands with reserved vinegar mixture and gently press 3/4 cup rice evenly across nori, leaving a 1/2-inch border on all sides. Spread 2 teaspoons mayonnaise evenly across center of rice. Layer 1/3 cup vegetable mixture and a few slices of avocado evenly across middle of rice. Sprinkle with sesame seeds. Starting with the edge closest to you, use the mat to fold the bottom third toward the center, pressing evenly across the roll to compress the filling. Repeat with another turn and gently press again. Remove mat. Lightly wet a sharp knife with vinegar mixture and cut into 6 equal pieces. Repeat with remaining sheets of nori. Serve with reserved pickled ginger and soy sauce.
If you don't have a bamboo mat, you can line a clean kitchen towel with plastic wrap and use that as a guide for rolling.