Preheat the oven to 350°. In a medium skillet, heat 2 tablespoons of the olive oil. Add the mushrooms and all but 2 tablespoons of the sliced scallions, and season with salt and pepper. Cook over moderate heat, stirring occasionally, until any liquid from the mushrooms has evaporated and the mushrooms are browned, about 8 minutes. Add the garlic and the remaining 1/2 tablespoon of olive oil and cook over moderately high heat, stirring, until the garlic is golden, about 2 minutes. Stir in the soy sauce and rice, then add the stock and water and bring to a boil. Add a large pinch each of salt and pepper.
Scrape the rice into a 9-inch square glass or ceramic baking dish. Cover the rice with foil and bake for 45 minutes, or until the liquid has been absorbed and the rice is tender. Let the pilaf stand, covered, for 10 minutes.
Squeeze the lime over the pilaf and fluff with a fork. Season with salt, garnish with the reserved 2 tablespoons of sliced scallions and serve.
The pilaf can be prepared through Step 2 and kept in a warm place, covered, for up to 30 minutes.