In 1 very large or 2 large nonreactive saucepans, combine the remaining scallions, garlic 1/3 cup of dill, 5 tablespoons of oil and 1/2 teaspoon of salt. Arrange the stuffed artichokes on top. Gently pour 3 cups of water over and around the artichokes and bring to a boil over high heat. Cover, reduce the heat to moderately low and cook, spooning the juices over the rice occasionally, until the rice is tender and a leaf can easily be pulled from an artichoke, about 50 minutes. Add 2 cups of hot water twice during cooking. Transfer the artichokes to a platter, spoon the sauce around them and serve warm or at room temperature.