In a large bowl, whisk the oil, Parmigiano cheese, lemon juice, salt and pepper to blend. Set the lemon sauce aside. (The sauce can be made up to 8 hours ahead. Cover and refrigerate. Bring to room temperature before using.)
Meanwhile, bring a large pot of salted water to a boil. Add the spaghetti and cook, stirring occasionally, until tender but still firm to the bite, about 8 minutes.
Using a pasta spider or tongs, transfer the pasta directly to the bowl of lemon sauce. Add 1 cup of pasta water total in ¼-cup increments as needed, and toss together until a creamy sauce forms. Add the basil and lemon zest and mix. Season the pasta with more salt to taste, if needed. Transfer to bowls and serve.
