Place mushrooms in medium bowl; add enough boiling water to cover. Let stand until mushrooms soften, about 45 minutes. Drain mushrooms. Cut off stems and discard; thinly slice caps. Heat peanut oil in heavy large wok or nonstick skillet over medium-high heat. Add cabbage, ginger, garlic, and mushrooms. Stir-fry until cabbage wilts, about 2 minutes. Add 2-inch green onion pieces; toss until green tops begin to wilt, about 30 seconds. Remove from heat. Mix in 1 tablespoon soy sauce. Cook noodles in large pot of boiling salted water until just tender but still firm to bite. Drain well; place in large bowl. Whisk sesame oil, next 3 ingredients, and 2 tablespoons soy sauce in small bowl. Add to noodles. Add sliced eggs, 3/4 cup cilantro, and cabbage mixture; toss to blend well. Season with salt and pepper. (Can be made 1 day ahead. Cover and chill, tossing occasionally.) Sprinkle salad with 2 chopped green onions, chopped egg, and remaining 1/4 cup cilantro and serve.