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 1 3/4 cup cooking water, drain pasta into colander and set aside.
Heat 1 tablesppon oil in now empy 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 water and heavy cream into pot, bringing to simmer and cook for 2 minutes. Remove pot from heat, stir in pasta and stir until coated. Stir in remaining 3 tablespoons of 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 cooking pasta water. Stir in basil and season with salt and pepper to taste.
You can add additional oil and cheese and serve.