Bring a 2-quart pot of water to a boil. Heat 4 inches of vegetable oil in a Dutch oven over medium heat until it reaches 300 degrees F on a deep-fry thermometer. Line a plate with paper towels.
Using a peeler, peel the eggplant lengthwise skipping every other stripe so the eggplant looks like a zebra pattern. Dice the eggplant into 1-inch cubes, place in a strainer and lightly sprinkle with salt. Let the water drain from the eggplant in the strainer to get rid of any bitterness, about 15 minutes. Discard the water and pat down the eggplant with paper towels.
Blanch the eggplant in the hot oil for 7 to 8 minutes (you only want the eggplant to get soft). Take the eggplant out of the oil to the prepared plate, and then turn up the temperature to 350 degrees F.
Once the temperature reaches 360 degrees F, shut off the heat. Then add the eggplant and turn the heat back on (this will help if the oil spills over a little bit). Deep fry the eggplant until deep golden brown in color, 2 to 3 minutes (the eggplant should look a little wrinkly and have a firm texture on the outside and be soft and silky on the inside). Place the eggplant on fresh paper towels to drain the oil and put aside.
Add the olive oil to an 8-inch saute pan over medium-low heat. Season the pasta water with salt, place the pasta in the boiling water and cook it 75 percent of the way (about 12 minutes).
Once the oil is warm, add the garlic and infuse the oil, about 2 minutes (the garlic should not sizzle or brown). Add the dried oregano, cook 1 more minute, and then add the tomato sauce and 1/4 cup pasta water. Reduce while the pasta finishes cooking. If it gets too dry, add more pasta water. Remove the garlic and add a pinch of salt and the torn basil.
Transfer the pasta to the tomato sauce along with some of the pasta cooking water (1/3 to 1/2 cup). Add the fried eggplant to the pan and toss to combine. Turn off the heat and add the Parmesan; toss again. Finish with the butter and toss a final time.
Place the pasta into plates and grate the ricotta salata with a large grater over top. Top with a basil leaf or two and serve immediately.
I like to bake my ricotta to dry it out and give it some color, so I brush it with some olive oil and bake it in a 325 degree F oven for 10 to 15 minutes.