Place the flour in a large bowl and make a well in the center. Pour in the olive oil and ½ cup water and, using a fork, slowly stir the flour into the wet ingredients until a dough forms. Transfer the dough to a clean work surface and knead until smooth and elastic, about 10 minutes. Wrap the dough in plastic wrap and let rest for 1 hour.
Unwrap the pasta dough and cut into 4 pieces. Working with 1 piece at a time, feed the dough through a pasta machine, slowly reducing the thickness to the penultimate setting, then cut the dough into ¼-inch-wide strips. Arrange a wooden skewer parallel to you then position 1 pasta strip underneath the skewer at a 45° angle. Press one end to the skewer and while rolling the skewer toward you with one hand, press the strip to the skewer with the other hand, conforming the pasta to the shape of the skewer into a tight coil. Slide the pasta coil off the skewer, repeat with the remaining strips, and let dry for 1 hour.
Meanwhile, quarter the eggplant lengthwise, then cut each quarter crosswise into ½-inch-thick slices. Place the eggplant in a colander, season liberally with salt, and toss to combine. Let the eggplant stand for 30 minutes, then rinse under cold running water until all the salt is gone. Squeeze all the moisture from the eggplant, then dry thoroughly on paper towels. In a large skillet, warm the olive oil over high heat. Working in batches, add the eggplant to the oil and fry, tossing occasionally, until dark golden brown all over, about 10 minutes. Using a slotted spoon, transfer the eggplant to paper towels to drain and discard all but 1 tablespoon of the olive oil.
On a cutting board, split each tomato open, remove and discard its seeds, then crush by hand into a bowl. Return the skillet to medium heat, stir in the thyme and garlic, and cook until fragrant, about 30 seconds. Add the tomatoes and cook, stirring, until warmed through, about 2 minutes. Remove the skillet from the heat and transfer the tomatoes to a bowl. Stir in the almonds, basil, and ¼ cup extra-virgin olive oil, then season the pesto with salt and pepper.
Warm the remaining 1 tablespoon extra-virgin olive oil in a small skillet. Add the bread crumbs and cook, tossing, until golden brown, about 4 minutes. Transfer the bread crumbs to a bowl and let cool. In a large pot of boiling salted water, cook the pasta until al dente, about 3 minutes. Drain the pasta, reserving ⅔ cup pasta water, then stir the pasta and water into the pesto until evenly combined. Divide the eggplant among 4 serving plates, spoon over the pasta and pesto, and sprinkle with bread crumbs and more basil before serving.