Bring the broth and water to a boil in saucepan over medium heat. Reduce the heat to low and keep hot.
In another saucepan, heat oil over medium heat. Add the scallions and cook for about 3 minutes. Stir in the rice, season with salt and pepper, to taste, and cook for another 2 minutes. Pour in the hot broth mixture and bring to a boil. Reduce the heat to a simmer and cover the saucepan. Cook until the liquid is absorbed and the rice is tender, about 20 minutes. Stir in walnuts and taste for seasoning, adjusting as necessary. Transfer the pilaf to a serving bowl and serve.