Make the Filling:
Lay the block of tofu flat on a cutting board and slice it lengthwise into four even pieces. Flip each piece onto its wide side and slice again into four even pieces to get a total of 16 long (about 4-1/2-inch) rectangular strips. Lay the tofu strips on a paper towel and pat them dry. In a small (about 11x7-inch) baking dish, combine the soy sauce, ginger, sugar, sesame oil, and salt. Stir until the sugar and salt dissolve. Add the tofu strips and carefully turn them until they’re evenly coated with the marinade. Let the tofu stand at room temperature for 15 to 20 minutes, turning once. (Or marinate the tofu for up to 2 hours, covered, in the refrigerator.)
Gently transfer the tofu from the marinade to a platter, brushing off any ginger clinging to it. Blot the tofu dry with paper towels. Heat the vegetable oil in a medium (10-inch) nonstick skillet over medium-high heat. Carefully add half of the tofu strips. Cook until nicely browned on one side, 1 to 2 min., and then gently turn with a spatula to brown the opposite side, another 1 to 2 minutes. Return the cooked tofu to the platter and cook the remaining tofu in the same manner. Let the tofu cool to room temperature or refrigerate, covered, until ready to roll.
Assemble the Rolls:
Bring a medium saucepan of water to a rolling boil over high heat. Drop in the rice noodles. Remove the pan from the heat and let stand for 8 to 10 minutes, gently lifting and stirring the noodles now and then as they soften, to cook them evenly and to keep them from clumping. Drain the noodles in a colander and rinse them with cold water to stop the cooking. You should have about 4 cups cooked noodles.
Arrange the noodles, rice paper wrappers, lettuce, herbs, scallions, and cooled tofu around a large cutting board or tray set before you. Have a platter nearby for the finished rolls.
Fill a large skillet halfway with very warm water. Slide a sheet of rice paper into the water and press gently to submerge it until it becomes very pliable, 15 to 30 seconds. Remove the rice paper carefully, shaking gently to help excess water drain off, and lay it before you on a cutting board or tray. (If the water becomes too cool to soften the paper, reheat it briefly on the stove.)
Starting on the lowest third of the wrapper and working away from you, line up a narrow tangle of noodles (about 1/4 cup), a row of lettuce strips (about 1/4 cup), 5 to 7 good-size mint leaves, a row of cilantro or basil leaves, and a row of scallion strips. Leave about 1 inch of empty space along the wrapper's bottom and side edges.
Starting from the edge closest to you, roll the wrapper up and over the fillings. Stop after the first turn to tuck and compress everything snugly into the wrapper. Once the first turn is good and tight, fold the right and left sides of the wrapper in onto the roll, closing off the ends, as though making an envelope.
Just above the cylinder you've already rolled, lay one strip of tofu. Roll the wrapper tightly, all the way to the top, and press the seam closed. If the wrapper is too dry to stick, dip your finger in the water and run it along the paper to moisten and then press the seam closed. Set the roll, seam side down, on the platter. Continue to fill and roll up the rice paper sheets until you've made 16 rolls. (Don't let the finished rolls touch one another or they'll stick.) Serve immediately (or drape a damp dishtowel over the rolls, wrap the platter tightly in plastic, and serve within two hours.)
You can assemble summer rolls up to 2 hours in advance as long as you cover them with a damp dishtowel and wrap them well with plastic to keep the rice paper from drying out.