Toss the squash with salt in a colander and let it drain for 30 minutes. Pat the squash dry with paper towels and carefully wipe away any residual salt to ensure the vegetable sears rather than steams.

Heat a tablespoon of oil in a large nonstick skillet over high heat until it just begins to smoke. Cook half of the squash, stirring occasionally, until golden brown and slightly charred, about 5 to 7 minutes; transfer to a large plate and repeat with another tablespoon of oil and the remaining squash.

Heat another tablespoon of oil in the empty skillet over medium heat until shimmering. Add the garlic and pepper flakes and cook until fragrant.

Stir the cooked squash back into the skillet and cook briefly until heated through.

Bring 4 quarts of water to a boil in a large pot. Add the pasta and salt and cook, stirring often, until al dente. Reserve 1/2 cup of the cooking water, then drain the pasta and return it to the pot.

Add the squash mixture, tomatoes, basil, pine nuts, vinegar, and the remaining 2 tablespoons of oil to the pasta and toss to combine. Season with salt and pepper to taste, adding the reserved cooking water as needed to loosen the sauce so it coats every noodle.

Serve immediately with a dusting of Parmesan.