Cut eggplants lengthwise into 20 (1/4-inch-thick) slices total with a long serrated knife and transfer to a colander. Toss with 2 tablespoons sea salt, then let drain 1 hour.
While eggplant drains, heat 2 tablespoons oil in a 4- to 5-quart heavy pot over moderately high heat until hot but not smoking, then sauté onion and garlic, stirring, until golden, about 6 minutes. Add tomatoes with their juice and remaining 1/2 teaspoon sea salt, then reduce heat and simmer, uncovered, stirring occasionally, until thickened, 20 to 25 minutes. Stir in basil.
Gently squeeze eggplant to remove excess moisture (this helps reduce any bitter flavor) and pat dry. Heat 1/4 inch oil (about 2 cups) in a 12-inch heavy skillet over moderately high heat until it registers 360°F on thermometer, then fry eggplant in 4 batches, turning over once with tongs, until golden, about 4 minutes per batch. Transfer with tongs to paper towels to drain.
Cook pasta in a large pot of boiling salted water until al dente, then drain in colander and rinse under cold water to stop cooking. Drain well and toss pasta with 1 tablespoon oil.
Put oven rack in middle position and preheat oven to 350°F Spread 1/2 cup tomato sauce in bottom of a 13- by 9-inch glass or ceramic baking dish.
Line up 5 pasta strands parallel to one another on a work surface and cut crosswise into thirds. Put 1 fried eggplant slice on work surface and stack cut pasta (15 pieces) across 1 end of slice. Roll up eggplant, leaving ends of pasta exposed, and transfer to baking dish, seam side down. Make more rolls in same manner with remaining eggplant and pasta, transferring to dish (rolls will fit snugly). Spoon remaining sauce over eggplant, then cover dish loosely with foil. Bake until rolls are heated through, about 15 minutes.
• Tomato sauce, without basil, can be made 2 days ahead and cooled, uncovered, then chilled, covered. Stir in basil before proceeding.
• To take the temperature of a shallow amount of oil, put bulb in skillet and turn thermometer facedown, resting other end against rim of skillet. Check temperature frequently.
• Dish can be assembled (but not baked) 1 day ahead and chilled, covered (basil may discolor). Bring to room temperature to bake.