Tender carrots, cucumbers, and daikon radish fill these vegan sushi rolls; the lettuce and pea shoots make an edible garnish; pickled ginger, wasabi paste, and a low-sodium soy mixture accompany the rice dish.
Place the rice in a sieve, and rinse under cold running water until the water runs clear. Set the sieve over a bowl, and let the rice drain for 20 minutes.
Place 4 tablespoons rice-wine vinegar, the sugar, and the salt in a small nonreactive saucepan set over medium heat, and stir just until dissolved, about 3 minutes; set hot vinegar mixture aside.
Combine soy sauce with remaining 1 tablespoon plus 1 teaspoon rice-wine vinegar and mirin; set the soy mixture aside.
Place drained rice and 1 3/4 cups water in a medium saucepan; bring to a simmer. Cover, and cook over very low heat until rice is tender and the water has evaporated, about 15 minutes. Remove the pan from heat, and let rice rest, covered, 20 minutes.
Transfer the rice to a large bowl, and toss with 2 tablespoons reserved vinegar mixture. Place a clean, damp kitchen towel over the bowl, and set aside.
Prepare an ice bath. Bring a medium saucepan of water to a boil. Add the carrots, and cook until just tender, about 1 1/2 minutes. Using a slotted spoon, transfer the carrots to the ice bath to cool. Dry on paper towels, and transfer to another large bowl. Add the cucumber and remaining vinegar mixture to carrots; toss to combine.
Using a pair of tongs, hold one sheet of nori over the flame of a burner until the nori is lightly toasted, about 30 seconds. Place the nori on a bamboo sushi mat with the longer side toward you. Dampen your fingers, and cover the nori with 1 cup reserved rice, leaving a 1/2-inch border at the edge nearest to you. Sprinkle 1/2 tablespoon sesame seeds over rice, cover with a second bamboo mat, and turn over so rice side is down. Remove top mat from nori.
Spread 1/4 teaspoon wasabi across the lower third of the nori. Arrange 2 carrots, 2 cucumbers, one-quarter of the daikon, 4 lettuce leaves, and one-quarter of the pea shoots in a line across the bottom third of the nori so that the vegetables extend beyond the edges. Using the bamboo mat, roll into a cylinder, enclosing the vegetables. Squeeze, sealing the cylinder, and remove the mat. Using a serrated knife, slice the cylinder into six equal pieces.
Repeat steps seven and eight with the remaining ingredients, making three more cylinders. Serve sushi with the remaining wasabi, the reserved soy mixture, and the pickled ginger, if desired; garnish with shiso leaves, if desired.
If you don't have bamboo sushi mats, you can use two pieces of plastic wrap instead.