Put a baking sheet in the upper and lower thirds of the oven and preheat to 450 degrees F. Put the tofu cubes on a clean kitchen towel in a single layer; let drain. Make the dressing: Whisk the rice vinegar, soy sauce, pickled ginger brine and 1/4 cup vegetable oil in a large bowl. Set aside.
Toss the mushrooms and squash with the remaining 2 tablespoons vegetable oil, a pinch of salt and a few grinds of pepper in a separate large bowl; spread the vegetables on the top hot baking sheet. Pat the tofu dry and add to the same bowl along with 3 tablespoons of the dressing; toss, then spread on the bottom hot baking sheet. Roast, switching the baking sheets and tossing the vegetables and tofu halfway through, until the squash is tender and the tofu is golden, 18 to 20 minutes.
Add the greens, snow peas, cashews and chopped pickled ginger to the bowl with the remaining dressing; toss. Divide among bowls; top with the roasted vegetables, tofu and sprouts.