Bring a large pot of salted water to a boil and cook the spaghetti until al dente. Reserve some pasta water before draining.
Meanwhile, heat the butter and olive oil in a large skillet over medium heat. Add the lemon zest and lemon juice and cook for about 1 minute. Pour in the heavy cream and season with salt to taste. Bring to a gentle simmer.
Add the cooked spaghetti and basil leaves directly to the skillet and toss until well coated.
Turn off the heat. Add the Parmigiano Reggiano and a splash of pasta water. Toss vigorously until creamy.
Serve immediately.