Bring a large pot of water to a boil. Add pasta and cook according to package instructions until al dente (cooked but firm). Reserve 1 cup of the pasta cooking water and drain the pasta. Return the pasta to the pot.
Meanwhile, combine oil and lemon zest in a small saucepan. Cook over low heat, stirring occasionally, until the zest starts to sizzle gently, about 5 minutes. Stir in cream and salt and increase heat to medium. Cook, stirring, until the cream is heated through and just beginning to simmer, about 2 minutes. Gradually whisk in 2 tablespoons lemon juice, 1 tablespoon at a time. Simmer until thickened, about 1 minute. Taste and add more lemon juice to taste, if desired. Remove from heat and cover to keep warm.
Add the sauce and ½ cup of the reserved cooking water to the cooked pasta. Toss gently to combine. Add 1 cup Parmigiano, basil, parsley, mint and pepper to taste; toss to coat the pasta evenly with the sauce, adding more of the cooking water as needed to loosen the sauce.
Serve topped with the remaining ½ cup Parmigiano.
