In a large saucepan, melt the butter in the oil. Add the onion and cook over moderate heat, stirring occasionally, until softened, about 5 minutes. Add the rice and cook, stirring, for 2 minutes. Add the broth and a pinch of salt and bring to a boil. Cover and simmer over very low heat until the broth is absorbed and the rice is al dente, 45 to 50 minutes. Remove from the heat and let the rice stand, covered, for 10 minutes. Fluff with a fork.
In a skillet, melt the remaining 2 tablespoons of butter. Add the mushrooms and cook over moderately high heat until their liquid is evaporated and the mushrooms are browned, about 8 minutes. Season the mushrooms with salt, stir them into the rice and serve.