Cook pasta according to the package directions. Reserve 1/2 cup of the cooking water; drain 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 1/4 teaspoon each salt and pepper to the skillet. Cook, tossing ofte, 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, parmesan cheese, 1/4 cup of the reserved cooking water, and 1/2 teaspoon salt to the pasta and toss gently to coat (add more cooking water if the pasta seems dry). Serve sprinkled with the chives and additional shredded parmesan cheese.
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