Spoon the ricotta into a large, fine-mesh sieve or a colander lined with a double thickness of cheesecloth or a basket-type coffee filter. Set the sieve over a bowl and cover the ricotta well with plastic wrap. Drain the ricotta in the refrigerator at least overnight, or up to 24 hours. Discard the liquid in the bottom of the bowl.

Make the tomato sauce.

Trim the stems and ends from the eggplants. Remove strips of peel about 1-inch wide from the eggplants, leaving about half the peel intact. Cut the eggplant lengthwise into1/4-inch thick slices and place them in a colander. Sprinkle generously with the coarse salt, tossing to expose all slices, and let drain for 1 hour. Rinse the eggplant under cool running water, drain thoroughly and pat dry.

Pour ½ cup each of the olive and vegetable oils into a medium skillet over medium-high heat. While the oil is heating, whisk 2 of the eggs and 1 teaspoon salt together in a wide, shallow bowl. Spread the flour in an even layer in a separate wide, shallow bowl or over a sheet of wax paper. Dredge the eggplant slices in flour, shaking the excess off. Dip the floured eggplant into the egg mixture, turning well to coat both sides evenly. Let excess egg drip back into the pan.

When a corner of a coated eggplant slice gives off a lively sizzle when dipped into the oil, it is ready for frying. Add as many of the coated eggplant slices as fit without touching and cook, turning once, until golden on both sides, about 4 minutes. Remove the eggplant to a baking pan lined with paper towel and repeat with the remaining eggplant slices. Adjust the heat as the eggplant cooks to prevent the egg coating cooking too fast or overbrowning. Add oil to the pan as necessary during cooking to keep the level more or less the same. Allow the oil to heat before adding more eggplant slices.

Preheat the oven to 375 degrees F. Stir the drained ricotta, 2/3 cup of the grated cheese and the parsley together in a mixing bowl. Taste and season with salt and pepper. Beat the remaining egg in a separate small bowl and stir it into the ricotta mixture. Pour 1 cup of the tomato sauce over the bottom of a 10x 15-inch baking dish. Sprinkle lightly with 2 tablespoons of the remaining grated Parmigiano-Reggiano cheese. (If you don’t have a baking dish of that size, divide the sauce and rolls between two smaller dishes into which they fit comfortably.)

Lay one of the fried eggplant slices in front of you with the short ends towards you. Spoon about 2 tablespoons of the ricotta filling over the narrow end of the slice and top it with a mozzarella stick. Roll into a compact roll and place, seam side down, in the prepared baking dish. Repeat with the remaining eggplant slices and filling, placing the rolls side by side.

Ladle the remaining tomato sauce over the eggplant rolls to coat them evenly. Sprinkle the remaining grated Parmigiano-Reggiano cheese over the top of the eggplant and tear the basil leaves, if using, over the cheese.

Cover the dish loosely with aluminum foil and bake until the edges of the casserole are bubbling and the filling is heated through, about 30 minutes. Let rest 10 minutes before serving.