Braised artichokes, filled with savory garlic-and-dill-flavored rice, are another Kotsoni family favorite. Medium artichokes are the perfect size for meze; don't use large one, which are unwieldly to clean and cook.
Using a serrated knife, trim the stems and 1 1/2 inches from the top of 1 artichoke. Remove the small outer leaves and a layer of the larger ones. Using a melon baller or spoon, scoop out the center cone of leaves and the furry choke and drop the artichoke into a very large bowl of water mixed with the lemon juice. Repeat with the remaining artichokes.
In a medium bowl, combine the rice, half of the scallions and garlic, the 1/2 cup of dill, 3 tablespoons of the olive oil, 1 1/2 teaspoons of salt and the pepper. Drain the artichokes well and loosely stuff the centers with the filling; the rice will expand during cooking.
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.