Bring a large pot of salted water to a boil. Add the noodles and broccoli slaw; cook, stirring until the noodles are al dente and the slaw is tender, about 3 minutes. Drain and rinse under cold water; then transfer to a large bowl. Add the sesame oil and season with salt - toss.
Meanwhile, spread out the tofu on a kitchen towel; lay another towel on top and press to remove as much water as possible. Heat 2 tablespoons oil in a Dutch oven or large deep skillet over high heat. Add the tofu and cook, turning, until golden, about 6 minutes. Transfer to a paper towel-lined plate using a slotted spoon; season with salt.
Heat the remaining 2 tablespoons oil in the pot. Add the scallions and ginger; stir-fry 30 seconds. Add the shitakes; stir-fry 3 minutes. Add the broth, soy sauce and tofu; season with pepper. Cook until the liquid is reduced by half, 3-4 minutes.
Whisk 3 tablespoons water and the cornstarch in a bowl; add to the pot and cook, stirring until thickened. Add the noodles and toss to coat.