Cook the pasta according to the package directions. Reserve ½ cup of the cooking water; drain the pasta and return it to the pot.
Meanwhile, heat the oil in a large skillet over medium heat. Add the shallot and cook, stirring occasionally, until soft, 3 to 4 minutes.
Add the squash and ¼ teaspoon each salt and pepper and cook, tossing often, until the vegetables are just beginning to soften, 4 to 5 minutes.
Add the cream and lemon zest and cook until the vegetables are tender and the cream is slightly thickened, 1 to 2 minutes.
Add the squash mixture, pecorino, ¼ cup of the reserved cooking water, and ½ teaspoon salt to the pasta and toss gently to combine (add more cooking water as needed to loosen the sauce).
Serve sprinkled with the chives and additional pecorino.