Cover mushrooms with 1 1/2 cups hot water in a medium bowl. Set a small bowl on top of mushrooms to keep them submerged. Let stand until mushrooms are soft, about 20 minutes. Remove mushrooms with a slotted spoon, and transfer to a work surface. Cut mushrooms crosswise into 1/4-inch strips; set aside. Strain soaking liquid; set aside.
Make the dipping sauce: Stir together 1/3 cup soaking liquid, the lime juice, 1 teaspoon tamari, 1 teaspoon cilantro, and the salt in a small bowl; top with some julienned scallions. Set aside.
Bring a medium pot of water to a boil. Add noodles; cook according to package instructions. Drain. Rinse with cold water; drain again. Transfer to a medium bowl; toss with sesame oil and remaining 2 tablespoons cilantro. Set aside.
Heat vegetable oil in a large skillet over medium-high heat until hot but not smoking. Stir in shallots, garlic, and ginger. Cook, stirring occasionally, until shallots begin to brown, about 2 minutes (if the shallots begin to stick, stir in 1 tablespoon reserved soaking liquid). Add cabbage, and toss to combine. Reduce heat to medium. Cook, stirring occasionally, until cabbage wilts slightly, 2 minutes. Stir in mushrooms, 1/4 cup reserved soaking liquid, and remaining 1 tablespoon plus 2 teaspoons tamari. Cover, and cook, stirring occasionally, until cabbage is tender, about 7 minutes. Stir in sliced scallions; set aside.
Toast the nori sheets: Using tongs, pass each sheet just over a medium gas flame, turning, until fragrant and crisp, about 3 passes on each side. (If only an electric burner is available, heat to medium; toast sheet directly on the burner, flipping for 30 seconds to 1 minute.)
Working with 1 toasted sheet at a time, put 1/6 of the noodles along 1 edge of the nori, leaving a 1-inch border. Top with 1/6 of the cabbage mixture. Starting at the side with the filling, roll up sheet, enclosing filling. Repeat process with remaining sheets, noodles, and cabbage mixture. Cut each roll on the diagonal into 2 pieces. Serve with dipping sauce.