Make the Dressing:** In a small bowl, whisk together the soy sauce, rice vinegar, brown sugar, and sesame oil until the sugar dissolves.
Set aside.
Prep the Salad:** Cut the baked tofu, celery, carrots, turnip, and cucumber into thin matchsticks. Thinly slice the shiitake mushroom caps and cut the green onions into ribbons.
Place everything in a large bowl along with the chopped cilantro.
Toss and Marinate:** Pour the dressing over the tofu and vegetable mixture.
Toss gently to coat.
Let the salad stand for at least 30 minutes, stirring occasionally, to allow the flavors to meld.
Assemble and Serve:** Just before serving, stir in the red pepper flakes.
Arrange the shredded romaine or baby kale on a platter or in bowls.
Top with the marinated salad mixture and garnish with toasted cashews and sesame seeds, if using.