Toss the squash with salt in a colander and let it drain for 30 minutes to remove excess water. Pat the pieces thoroughly dry with paper towels and wipe away any residual salt to ensure proper browning.

Heat a portion of the oil in a large nonstick skillet over high heat until it just begins to smoke. Sear the squash in two separate batches, cooking until golden brown and slightly charred, about 5 to 7 minutes per batch; reduce the heat if the pan begins to scorch.

Add more oil to the empty skillet and reduce the heat to medium. Sauté the shallots until softened and lightly browned, about 2 to 4 minutes.

Stir in the capers, lemon zest, pepper, and the cooked squash, and cook for about 30 seconds to allow the flavors to combine.

Boil the pasta in 4 quarts of water with the remaining salt until al dente. Reserve a half-cup of the starchy cooking water before draining the pasta.

Return the pasta to the pot and toss with the squash mixture, lemon juice, tomatoes, parsley, and the remaining oil. Add the reserved cooking water as needed to loosen the consistency, then serve topped with crumbled goat cheese.