Bring a pot of salted water to boil. Grate parmesan, then zest and juice one lemon (separating both). Set aside. Add spaghetti to boiling water and start making the sauce.
Add oil, butter and garlic to a saucepan and sauté over medium heat. Next, add heavy cream and bring to a simmer and whisk. Add pasta water, then stir in ¾ of lemon zest and stir.
Add half of parmesan. Stir constantly until parmesan evenly dissolves. Add lemon juice, stir, then add remaining parmesan and stir again.
Once pasta is cooked and sauce has slightly thickened, remove from heat and divide among bowls. Top with remaining lemon zest, fresh parmesan and a grind of black pepper.