Bring a large pot of water to boil over high heat. Add the noodles and cook until al dente, about 8 minutes. Drain and divide them among 4 large soup bowls.
In a medium saucepan over medium heat, combine 3 cups water with the soy sauce, mirin, vinegar, and ginger and bring it to a simmer. Add the snow peas, mushrooms, and tofu to the pan. Simmer until the snow peas are crisp-tender and the tofu is heated through, about 3 to 5 minutes.
Divide the lettuce, radishes, carrots, and scallions among the bowls. Ladle the hot soup over the noodles and vegetables. Drizzle each serving with a few drops of sesame oil and hot sauce, if desired. Serve immediately.