How to Make It
Combine tomatoes and onions in a large stockpot over medium. Cook, stirring constantly, until tomatoes begin to release their juices, about 5 minutes. Cover and cook, stirring occasionally, until tomatoes are fully broken down and very soft, about 30 minutes.
Using a spider or slotted spoon, transfer tomatoes and onions to a large heatproof bowl; reserve liquid in pot. Place a food mill over stockpot. Pass tomatoes and onions through food mill into stockpot, discarding tomato skins and other solids. Stir in fine sea salt, and reduce heat to medium-low. Cover partially, and cook, stirring occasionally, until thickened and reduced to 7 cups, about 1 hour and 15 minutes. Stir in basil leaves; cook 5 more minutes. Remove from heat, and cover to keep warm.
While tomato sauce cooks, heat corn oil in a high-sided skillet or large Dutch oven over medium-high, about 5 minutes. Add 3 to 4 cups eggplant to skillet in a single layer. (Eggplant should be almost fully submerged in oil.) Sprinkle with 1 teaspoon non-iodized salt. Fry, stirring occasionally, until eggplant cubes are golden brown and soft to the touch with a crispy exterior, 8 to 10 minutes. Using a spider or slotted spoon, transfer eggplant to a large plate to drain; reserve drained oil. Repeat with a second batch of eggplant and 1/2 teaspoon non-iodized salt. Repeat with a third batch of eggplant and 1/4 teaspoon non-iodized salt. Fry remaining eggplant, in batches if needed, adding 1/8 teaspoon non-iodized salt to each batch.
Cook pasta in a large pot of salted water according to package directions for al dente. Drain. Add pasta to tomato sauce, and stir to coat. Fold in fried eggplant and drained oil.
To serve, transfer pasta to a rimmed platter or large shallow bowl. Top with ricotta salata, and garnish with fresh basil leaves.