Bring 4 quarts water to boil in large Dutch oven over high heat. Add 1 tablespoon salt and pasta to boiling water; cook, stirring frequently, until al dente. Reserve 1¾ cups cooking water, drain pasta into colander, and set aside.

Heat 1 tablespoon oil in now-empty Dutch oven over medium heat until shimmering. Add shallot and 1/2 teaspoon salt; cook until shallot is softened, about 2 minutes. Whisk 1 1/2 cups of reserved pasta cooking water and cream into pot; bring to simmer and cook for 2 minutes. Remove pot from heat, return pasta, and stir until coated. Stir in remaining 3 tablespoons oil, lemon zest, lemon juice, 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.

Let the dish rest briefly before serving so the flavors develop and the sauce thickens.