Step 1

Heat oil in a large skillet over medium-high. Arrange squash in skillet in a single layer and cook undisturbed until browned, about 5 minutes. Shake skillet, scraping bottom with a spatula to release squash, and turn pieces. Add garlic, soppressata, and thyme sprigs; season with salt and pepper. Cook, tossing occasionally, until squash is tender and garlic is browned all over, about 5 minutes. Remove from heat; set aside.

Step 2

Cook pasta in a large pot of boiling salted water, tossing occasionally, until very al dente, about 7 minutes.

Step 3

Using tongs, transfer pasta to reserved squash mixture and add 1½ cups pasta cooking liquid. Set over high heat; cook, tossing, until pasta is al dente and liquid is reduced by half, about 3 minutes. Reduce heat to medium-low and gradually add 3 oz. Parmesan, tossing until cheese is melted and pasta is coated in a smooth, glossy sauce. Thin with more pasta cooking liquid if needed. Pluck out thyme sprigs.

Step 4

Divide pasta among bowls and top with thyme leaves and more Parmesan.