For the Marinara Sauce:
Pour the tomatoes and purée into a bowl and crush with your hands or the back of a spoon, then set aside.
Add the extra-virgin olive oil and garlic cloves in a large skillet, and bring to medium heat. As the garlic begins to sizzle, stir occasionally, and cook until fragrant and golden, about 5 minutes. Monitor the heat so the garlic doesn't cook too quickly or burn. Add the red pepper flakes and cook for another 30 seconds.
Remove the skillet from the heat and let the oil cool slightly. Carefully stir the tomato paste into the garlic and oil. Be EXTRA mindful so the oil doesn't splatter and burn you as the paste is added—and cook for 1-2 minutes. Spoon the tomatoes into the pan with the oil and paste and season with salt and pepper. Add the basil sprigs and bring to a boil. Reduce to a bubbling simmer over low heat, stirring occasionally, for 30 minutes. Stir to mix the oil that rises to the top, then discard the basil sprigs and garlic. Remove sauce from heat.
For the Pasta:
While the sauce is simmering, roast your eggplant. Preheat your oven to 425°F. Cut the ends off the eggplants. Slice them in half, vertically, then cut them into ½-inch (half-moon) shaped pieces. Brush a sheet pan with olive oil and arrange the eggplant pieces in an even layer. Use two sheet pans if needed. Brush the eggplant tops with olive oil, sprinkle with kosher salt, and place in the oven to roast until golden brown, for about 30 minutes or so.
While the eggplant roasts, fill a large pot with water and bring to a boil. Once boiling, salt the water generously. Add the pasta and cook until al dente. Use a mug or Pyrex to reserve ¾ cup of the pasta water. Drain, then add the pasta, basil, marinara, and roasted eggplant to the large pot you cooked the pasta in. Gradually stir in the pasta water (start with ½ cup, then add more if needed) and stir well to combine. Taste and add more salt or red pepper flakes if desired. Transfer the pasta to a serving bowl and top with freshly grated ricotta salata and more basil.