Bring a large pot of water to a boil. Salt the water, add the pasta and cook until al dente. Drain, reserving a ladle of the water.
While the pasta is working, in a large, deep skillet, heat the EVOO, 4 turns of the pan, the garlic and the crushed red pepper over medium-high heat. Add the eggplant, season with salt and cook until tender, 12 to 15 minutes. Using a potato masher or the back of a wooden spoon, break up the eggplant into a coarse, thick sauce. Stir in the tomatoes and heat through, then stir in the parsley and simmer for a couple of minutes. Season with salt and black pepper. Stir in the reserved pasta cooking water, add the pasta and toss.
To serve, divide the pasta among 4 shallow plates or bowls. Place 1/4 cup ricotta cheese alongside each serving (to mix into each bite of the pasta as you eat it). Sprinkle with the basil and arugula. Pass Parmigiano-Reggiano around the table.