To make the dressing, combine tofu, soy milk, ginger, soy sauce, miso and mustard in a blender or food processor. Process just until smooth and creamy. Transfer to a bowl and stir in the cilantro and green onion. Cover and refrigerate for at least 1 hour.
In a small bowl, toss the avocado slices in the lemon juice to prevent browning. Set aside. In a large bowl, combine the lettuces, red and green onions, and cilantro and toss to mix.
Add 2/3 of the dressing and toss lightly to coat. Divide the salad among individual plates. Arrange 2 avocado slices on top of each portion in a crisscross pattern. Top each avocado cross with a drizzle of the remaining dressing. Serve immediately.