Preheat the oven to 275◦F.
Prep all vegetables according to specifications above.
Roast the vegetables: Arrange the cut eggplant and tomatoes in a single layer on separate parchment-lined sheet pans. Do not overcrowd the pans. (Use 2 pans for each vegetable, if necessary.) Toss the veggies lightly with about half of the olive oil, then place on a middle oven rack for 2 hours at 275◦F, stirring once about halfway through the cooking process.
Once the vegetables are finished, remove them from the oven and sprinkle them lightly with salt. Set aside. (NOTE: The vegetables can be done several hours or up to 1 day in advance. If doing so, hold the roasted vegetables in a sealed container under refrigeration until it is time to cook the pasta.)
Bring a large pot of water to a boil. Once the water comes to a boil, add salt, then the pasta, stirring frequently. Cook the pasta for 2 minutes less than the package directions, or about 2 minutes before you think it is al dente. (Be sure to reserve at least 1 cup of the starchy pasta water.)
Meanwhile, while the pasta is cooking, combine the garlic and remaining olive oil in a large skillet and heat on medium-high. Sauté for 2-3 minutes or until just starting to brown, then add the roasted vegetables to the pan, just to heat through.
When the pasta is ready, transfer it to the skillet along with about ½ cup of the starchy pasta water (to start) and the ricotta. Stir well to fully combine all ingredients and let the pasta finish cooking in the sauce and absorb all of those wonderful flavors! Add additional pasta water if more moisture is needed.
When the pasta is al dente, turn off the heat, then fold in the pecorino and basil, and season to taste with salt and pepper. Finish with a healthy drizzle of extra-virgin olive oil.
Serve warm with additional fresh basil. Buon Appetito!