Bring 4 quarts of water to a boil in a large dutch oven. Add 1 tablespoon salt and the pasta to the boiling water and cook until al dente. Reserve 1 3/4 cups cooking water, drain pasta into colander and set aside.
Heat 1 tablespoon oil in the now-empty pan and heat over medium heat until shimmering. Add the shallot and 1/2 teaspoon salt and cook until softened, about 2 minutes. Whisk 1 1/2 cups of reserved past cooking water and cream into pot; bring to a simmer and cook for 2 minutes. Remove pot from heat, return pasta and stir until coated. Stir in remaining 3 tablespoons of oil, the lemon zest, juice, Parmesan cheese and 1/2 teaspoon pepper.
Cover and let pasta stand 2 minutes, tossing frequently and adjusting consistency with remaining 1/4 cup reserved pasta water, if necessary. Stir in basil and season with salt and pepper to taste. Serve, drizzling individual portions with oil and sprinkling with cheese.