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—removing the moisture is key for a proper sear.

Heat oil in a 12-inch nonstick skillet over high heat until just smoking. Add half of the squash and cook, stirring occasionally, until golden brown and slightly charred, about 5 to 7 minutes; transfer to a large plate and repeat the process with more oil and the remaining squash.

Heat oil in the now-empty skillet over medium heat until shimmering. Stir in the garlic and pepper flakes and cook until fragrant, then add the cooked squash back to the pan and toss until heated through.

Bring a large pot of water to a boil, add the pasta and salt, and cook until al dente. Reserve half a cup of the cooking water before draining the pasta and returning it to the pot.

Add the squash mixture, tomatoes, basil, pine nuts, vinegar, and the remaining oil to the pot with the pasta and toss to combine. Season with salt and pepper to taste, adding the reserved cooking water as needed to loosen the consistency.

Serve immediately while the squash is warm and the basil is vibrant, with plenty of Parmesan on top.