Arrange eggplant slices in a single layer on wire racks set in rimmed baking sheets. Sprinkle lightly all over on both sides with salt, then let stand until eggplants look dewy with water and some water has dripped below, about 20 minutes. Blot well with clean towels to remove all excess water.

Meanwhile, in a small saucepan, heat olive oil with garlic clove over medium heat until garlic begins to sizzle. Reduce heat to medium low and continue to cook until garlic is just beginning to turn golden, about 4 minutes. Remove garlic clove and discard. Add blended canned tomatoes (or passata) along with the basil and bring to a simmer over medium-high heat. Reduce heat to maintain a gentle simmer and cook, stirring often, until thickened and no longer watery (this may take almost no time at all for some passata products, or up to 20 minutes for waterier canned tomatoes). Remove from heat and season with salt. Set aside.

In a 12-inch cast iron or stainless-steel skillet, heat about ¼ inch of oil over medium-high heat until shimmering. Working in batches, fry eggplant, turning once, until golden on both sides. Carefully transfer to paper towels to drain well, blotting to soak up any excess oil as you go. Repeat with remaining eggplant, adding more oil to skillet if necessary. Set aside.

Bring a large pot of salted water to a boil. Prepare an ice bath in a large mixing bowl. Working in batches to avoid crowing the pot, boil lasagna noodles until just shy of al dente, about 2 minutes less than cooking instructions for fresh lasagna noodles or 3 minutes less than package instructions for dry lasagna noodles. Using a strainer and tongs, carefully transfer to ice bath to chill, then drain and blot lightly with towels. Repeat with remaining pasta.

Preheat oven to 375°F (190°C) and set rack in middle position. In a medium mixing bowl, whisk ricotta until very smooth and creamy. Season with salt.

In a 2-quart baking dish, spoon a thin layer of tomato sauce all over, spreading it evenly to coat bottom. Cutting cooked lasagna noodles as needed to fit the baking dish, arrange the pasta in a single, even layer. Spoon another thin layer of sauce on top of the pasta, then dollop and gently spread a thin layer of ricotta all over (it doesn't need to cover every inch, but should be evenly enough distributed that it will melt and spread when baked). Scatter a small handful of mozzarella all over, then pieces of fried eggplant in a single, even layer. Top with another thin layer of sauce, then a sprinkle of grated ricotta salata and a dusting of Parmigiano-Reggiano. Repeat this layering until the baking dish is full, making sure to top the final layer of pasta only with tomato sauce, ricotta, ricotta salata, and Parmigiano-Reggiano. (Note: You may not use up all the components, but you should make sure to use all the eggplant and ricotta when layering the dish.)

Set baking dish on a rimmed baking sheet (to catch any drips) and bake until fully heated through and lasagna is very lightly browned on top, about 30 minutes. Let rest for 15 minutes to cool and settle, then slice and serve, sprinkling torn fresh basil leaves on top.