Preheat the oven to 425°F and line two large baking sheets with parchment paper.

Place the tomatoes and their juices in a large bowl and use your hands to crush them. Set aside.

Halve the eggplants lengthwise, then cut each half into 1/4- to 1/2-inch-thick half-moons. Cut any very large half-moons in half to form wedges.

Divide the eggplant between the two baking sheets. Drizzle each sheet with 2 tablespoons of the olive oil (4 tablespoons total) and sprinkle generously with salt and pepper. Toss to coat and spread evenly on the baking sheets. Roast for 25 to 35 minutes, or until tender and lightly browned.

Bring a large pot of salted water to a boil. Prepare the pasta according to the package instructions, cooking until al dente.

Meanwhile, heat the remaining 1 tablespoon olive oil in a large skillet over medium heat. Add the garlic, oregano, and red pepper flakes and sizzle for 30 seconds. Stir in the tomato paste, then add the crushed tomatoes, salt, and several grinds of pepper. Simmer for 10 minutes, or until slightly thickened.

When the pasta is al dente, use a spider or slotted spoon to transfer it from the pot to the skillet with the sauce. Add the roasted eggplant and stir to coat. Add the cheese and stir until melted, adding pasta water as needed to help it melt into the sauce.

Season to taste and serve topped with basil and more cheese, if desired.