Toss the squash with salt in a colander and set it over a bowl to drain for 30 minutes. Use this time to prep the remaining ingredients.

Pat the squash dry with paper towels to remove excess moisture. This ensures the squash will brown effectively when it hits the heat.

Boil the pasta in a large pot of salted water until al dente. Reserve 1/2 cup of the pasta cooking water, then drain the pasta and return it to the pot.

Heat 3 tablespoons of olive oil in a large skillet over medium-high heat until the oil should shimmer but not smoke. Add the squash in a single layer and cook until browned on the edges and just tender, about 8 to 10 minutes.

Transfer the squash to a plate. Add the remaining 2 tablespoons of oil to the skillet along with the shallots and capers. Cook until the shallots are softened and translucent, about 2 minutes.

Stir in the tomatoes, lemon zest, and lemon juice. Cook until the tomatoes just begin to soften and release their juice.

Combine the cooked pasta, squash mixture, parsley, half of the goat cheese, and the reserved pasta water in the pasta pot. Toss gently over low heat until the cheese melts into a light, creamy sauce that coats the noodles.

Season with pepper and serve immediately, topping each bowl with the remaining crumbled goat cheese.