Boil the spaghetti in well-salted water according to package directions.
While it boils, finely grate the zest of half your lemon into the large bowl you'll use to serve your dish.
Add the juice of the whole lemon (about 4 tablespoons).
Grate the parmesan on top.
Add olive oil, about ½ teaspoon salt (and more to taste), and several grinds of black pepper and combine them with a whisk until very well mixed and as smooth as possible.
When the pasta reaches the ideal texture, scoop out 1 cup of cooking water, set it aside, and drain the rest.
Quickly turn piping hot spaghetti to lemon-parmesan mixture in bowl and use tongs or spoons to toss it until all of the strands are coated.
Add reserved pasta water, a tiny splash (about 1 to 2 tablespoons) at a time, tossing the whole time, until the spaghetti looks glossy and lightly sauced.
Add basil leaves and toss to combine, then serve.
Finish each serving with a little extra olive oil, salt, pepper, and parmesan.
