In a large bowl, toss eggplant cubes with kosher salt and 4 tablespoons (¼ cup) of the olive oil. Spread onto prepared baking sheet in an even layer. Bake until very tender, 35 to 40 minutes, tossing halfway through baking time.
Meanwhile, heat remaining 2 tablespoons olive oil in a large skillet over medium heat. Add garlic and red pepper flakes; cook and stir until fragrant, about 1 minute. Add tomatoes; cook and stir until thickened, stirring occasionally and crushing tomatoes with a spoon (recipe note #1), 10-12 minutes. (If sauce thickens before eggplant finishes cooking, remove from heat. Return to heat in step 6 adding a minute or two to ensure that sauce is heated through.)
While tomato mixture cooks, bring a large pot of salted water to boil. Cook pasta until firm to the bite (al dente) according to package directions. Drain pasta, reserving 1 cup cooking water.
Add eggplant and pasta to tomato mixture; cook and stir 2 to 3 minutes until combined and pasta is well coated, adding reserved pasta water a little at a time if it seems too dry. Season to taste with salt. Remove from heat and stir in half of the basil. Transfer to platter or individual pasta bowls and top with remaining basil and grated cheese. Serve.
TIP: Crushing whole tomatoes in the pan can be messy. Instead, use kitchen shears to cut them into smaller pieces right in the can!