Season the tofu with salt and pepper. In a small bowl, combine the soy sauce, 1 tablespoon grated ginger and the sesame oil.
In a large pot, bring 4 quarts salted water and the remaining 2 tablespoons grated ginger to a boil. Add the pasta and cook until al dente; drain, reserving 2 cups pasta cooking water. Return the pasta to the pot.
Meanwhile, in a large nonstick skillet, heat the vegetable oil over medium-high heat. Add the tofu in a single layer and cook, undisturbed, until browned on 1 side, about 3 minutes. Repeat, turning the cubes, until browned on all sides; transfer to a plate. Add the mushrooms to the skillet and cook, stirring occasionally, until browned, about 5 minutes.
Add the broccoli slaw, mushrooms, ginger-soy mixture and reserved pasta cooking water to the pasta and toss to coat. Divide the pasta and broth among 4 bowls and top with the tofu.