Trim the stems from the eggplants and peel strips of the skin about 1-inch wide, leaving about half of the peel intact.
Cut the eggplant into 1-inch cubes and toss in a large bowl with the two tablespoons salt.
Place in a colander over a large bowl, or in the sink, and let stand for 30 to 45 minutes to drain.
Rinse the eggplant under cool running water, drain thoroughly, and pay dry.
Preheat oven to 425°F.
Brush a rimmed baking sheet with 3 tablespoons of the olive oil. Turn the eggplant cubes onto the baking sheet and toss to coat with oil, and spread them out in an even layer.
Bake until the eggplant is very tender and lightly browned, about 25 - 30 minutes. Turn and stir the eggplant gently once or twice to ensure even baking.
Heat the remaining 3 tablespoons olive oil in a large skillet over medium heat.
Add the onion and saute until tender, about 5 minutes.
Add the garlic and saute for another 30 seconds.
Add the tomatoes (crushed by hand) and the juices, along with the red pepper flakes and sugar to the pan. Season with a good pinch of Kosher and black pepper (1/2 teaspoon each).
Reduce the heat to low and simmer, uncovered, until the sauce has thickened, about 15 minutes.
Meanwhile, bring a large pot of salted water to a rolling boil.
Add the pasta, and cook until al dente, according to package directions (stirring occasionally).
Drain the pasta and return it to the pot over low heat.
Pour in about 1/2 of the sauce, tossing lightly to coat the pasta.
Remove the pot from the heat, stir in the ricotta salata and basil, then stir in the eggplant. Mix until just blended.
Plate the pasta and spoon the reserved sauce over each serving, then top with the Parmigiano-Reggiano and a small basil leaf for garnish.