Whisk the vinegar and sugar together in a small bowl and set aside.
Combine the rice, water, and 1 teaspoon salt in a small saucepan with a tight-fitting lid. Wrap the pan's lid tightly with a small kitchen towel and cover the saucepan. (Make sure the towel's edges are folded up well away from the heat.) Bring to a boil, lower the heat to low, and simmer, covered, for 20 minutes.
Remove the rice from the heat (don't uncover) and set aside for 10 minutes. Fluff the rice with a fork and transfer it to a large bowl. Add the vinegar mixture and toss with the rice, using a wooden spoon. Spread it out on a parchment paper-lined baking sheet. Fan the rice continuously with a fan or magazine until cooled. Cover the rice with a damp towel.
Pour about 1 inch of water into a large saucepan and set up a collapsible steamer inside. Bring the water to boil, put the carrots and mushrooms in the steamer, cover, and steam until tender, about 6 minutes. Transfer the carrots and mushrooms to a large bowl, add the remaining 2 teaspoons salt, sesame oil, and sesame seeds, and stir to combine. Let cool.
When the vegetable mixture has cooled, add the cucumber, pickled daikon, lime juice, and lime zest and toss to combine. Add the rice and, using a rubber spatula, lightly mix with the vegetable mixture.
Set a small bowl of water beside you. Lightly wet your hands and form the vegetable rice into 12 small football-like shapes. Stuff the rice into the tofu pockets and press the sides together. Stuff a few tuffs of radish sprouts in the sides, if using. Stand the tofu packages up on a serving platter and sprinkle with the sansho. Serve with pickled ginger, wasabi, and soy sauce.
Serving suggestions: Pickled ginger, wasabi, and soy sauce.
Sansho is Japanese-style Sichuan pepper, available at Asian and health food stores. It's a fragrant, lemony pepper that's literally zippy on the tongue, but without spicy heat. We use it to finish a variety of dishes, from soups to rice to meats.
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