You might wonder why we fry the eggplant until it gets deliciously crisp and then layer it with tomato sauce and cheese, which makes it soft. Frying is used here for the flavor it imparts, not for the texture.
Do not peel the eggplants. Cut them lengthwise into 1/8-inch-thick slices. A mandoline is a great tool for this. Salt for 20 minutes, then rinse and dry.
Pour the olive oil into a large skillet to a depth of 1 inch and place over medium-high heat until hot, about 350°F. Meanwhile, season the flour with the pepper (since the eggplant is salted, no more is necessary). Dredge the slices in the flour and fry, turning once, until browned, about 2 minutes total. Drain on paper towels. The oil should be hot enough to bubble gaily while the eggplant is cooking. Add more oil as necessary to maintain the oil at a depth of about 1 inch.
Preheat the oven to 375°F. Spread a thin layer of the tomato sauce in the bottom of a 2-1/2-quart baking dish. A shallow dish (about 2 inches deep) is best. Arrange a layer of eggplant in the dish to cover the bottom completely. Spoon on another layer of sauce, and sprinkle with 2 tablespoons of the Parmesan, 1 tablespoon of the parsley, and dot with one-third of the chabis. Repeat the layering twice, beginning with eggplant and ending with chabis. Do not press down on the layers. Top with the remaining eggplant and then another thin layer of sauce. Sprinkle with the remaining Parmesan (you should have about 3 tablespoons), parsley (about 2 tablespoons), and the breadcrumbs. (The dish can be made to this point a day ahead, covered and refrigerated. Return to room temperature before baking, or add another 10 minutes or so if baking straight from the refrigerator.)
Bake until browned and bubbling, about 30 minutes. Do not over-brown. Let rest a few minutes before cutting into squares to serve. Serve hot or warm, drizzled with the garlic or basil oil if desired
Chef's Note: You will be glad to have leftover lasagnette. Just reheat it in a slow oven or cover and microwave. If you have any leftover sauce, refrigerate it and use as an addition to soups, stews, or pasta sauce. If you have used a deeper baking dish, you will need to determine if the lasagnette is hot in the middle. Use an old chef's trick: Insert a knife or metal skewer into the center of the dish, then withdraw it and lay it on your tongue. If it is hot, the dish is done. (This method is recommended only for those cooks sensible enough not to burn their tongues!)
