Bring 6 quarts of water to a boil and add 2 tablespoons salt. Add the eggplants and boil 10 minutes. Dip the bread in the water and remove, squeezing to drain the water and set aside. Remove the eggplants from the water and drain well. When the eggplants have cooled, coarsely chop them and place them in the work bowl of a food processor. Add the soaked bread and 1 egg and pulse to form a somewhat homogenous paste adding more egg if necessary. Add 1/2 cup pecorino, garlic and parsley and continue to pulse until a smooth yet somewhat dry mixture is formed. Do not over-process.
Turn the mixture into a mixing bowl and place bread crumbs on a large plate. Using your hands and working quickly, roll the eggplant mixture into balls with a 1/2-inch diameter and roll each ball in the bread crumbs to coat. Hold each ball on a sheet tray.
In a heavy-bottomed skillet, heat the olive oil until just smoking. Cook the eggplant balls in the oil a few at a time, for 6 to 7 minutes, until dark golden brown, turning gently with a wooden spoon to evenly brown all sides. Remove to a tray or plate lined with paper towels. Sprinkle with the remaining pecorino and serve.