In a large skillet over medium-high heat, warm the vegetable oil. Add the green onions and ginger and stir-fry until aromatic, about 30 seconds. Add the mushrooms and stir until coated with oil, about 30 seconds. Add the broccoli rabe or Chinese broccoli and stir until heated through, about 1 minute. Cover and cook until the greens are just tender-crisp, stirring occasionally, about 3 minutes. Uncover, add the tofu and stir gently. Stir the broth mixture to dissolve the cornstarch, and then add to the skillet. Cook until the sauce thickens and the tofu is heated through, 2–3 minutes. Season to taste with salt. Spoon the rice onto warmed plates, top with the tofu mixture, and serve right away.