Method
Cook the pasta
Bring a large pot of well-salted water to a boil. Cook the pasta until 2 minutes shy of al dente. Reserve 200ml of pasta water before draining.
Make the sauce
In a wide pan over low heat, melt half the butter. Add the lemon zest and let it bloom for 30 seconds. Add a splash of pasta water and the lemon juice, and let it simmer gently for a minute.
Finish the pasta
Add the drained pasta to the pan. Toss well over low heat, adding pasta water a little at a time. Add the remaining cold butter cubes and keep tossing — the cold butter emulsifies into the sauce and makes it silky. Remove from heat, add the Parmigiano, toss again. The sauce should coat the pasta in a glossy, clingy film.
Season and serve
Taste — it should be bright and buttery with a definite lemon backbone. Season with flaky salt and black pepper. Serve immediately with extra Parmigiano and a few basil leaves if you have them.
On the lemon: Use unwaxed lemons and zest them generously. The zest does more work than the juice here — it carries the aromatic oils that make the dish sing. If your lemons are very acidic, start with the juice of one and taste before adding more.