In a large Dutch oven, bring 4 quarts of water to boil. Add 1 tablespoon salt and pasta to water. Cook, stirring frequently, until pasta is al dente. Reserve 1¾ cups of pasta water, drain pasta and set aside.

In the now-empty Dutch oven, heat 1 tablespoon of oil. Once hot, add shallot and ½ teaspoon salt, cooking until shallot has softened, about 2 minutes. Whisk in 1½ cups of the reserved pasta water and cream. Bring to a simmer and cook for 2 minutes. Remove from heat and add pasta, tossing to coat. Stir in remaining 3 tablespoons of oil, lemon zest and juice, cheese and ½ teaspoon of pepper.

Cover and allow pasta to stand for about 2 minutes, tossing halfway through, so that the sauce can thicken and pasta can absorb the sauce. If needed, add the remaining ¼ cup of pasta water to adjust the consistency of the sauce. Stir in basil and add salt and pepper to taste. To serve, drizzle each individual portion with olive oil and sprinkle with additional Parmesan cheese.