Toss the eggplant slices with 1 teaspoon salt in a large bowl, then transfer to the bowl of a salad spinner set in the sink and let stand for 30 minutes.

Meanwhile, heat 2 tablespoons of the olive oil and the butter in large saucepan over medium-high heat until the butter has melted and foaming subsides. Add the onion and cook, stirring frequently, until softened but not browned, about 3 minutes. Add the garlic, oregano, and pepper flakes and cook, stirring, until fragrant, about 1 minute. Add the tomato paste and stir until homogeneous, about 30 seconds. Add the tomatoes, with their juice, and the basil stem, if using. Bring to a boil over high heat, reduce to a simmer, and cook, stirring occasionally, until the sauce has thickened and reduced to 4 cups, about 30 minutes. Set aside.

Spin the eggplant slices in the salad spinner to remove excess water, then place on a double layer of paper towels. Place another layer of paper towels on top and press down to extract as much moisture as possible.

Heat the remaining ¼ cup extra-virgin olive oil in a large nonstick or cast-iron skillet over medium-low heat until shimmering. Add the eggplant slices in a single layer (you may have to work in batches or in two pans) and cook, turning them and shaking the pan occasionally, until a deep caramelized brown on both sides and completely tender in the center, 7 to 10 minutes. Transfer to a paper-towel-lined plate and immediately season with salt.

Place the pasta in a large pot and cover with hot water. Add a large pinch of salt and bring to a boil over high heat, stirring occasionally to keep the pasta from sticking. Continue to cook until the pasta is fully softened but retains a slight bite in the center. Drain the pasta, reserving 1½ cups of cooking liquid, and return to the pot.

Add the sauce and stir to combine, adding some of the reserved pasta water as necessary to thin the sauce to the desired consistency. Fold in the caramelized eggplant. Serve immediately, topped with the basil leaves, if you have them, and grated cheese.