Grate the zest and squeeze the juice from 2 of the lemons.
Using a sharp serrated or other knife, cut off the tops and bottoms of the remaining 2 lemons to expose the flesh. Stand each lemon upright on the cutting board and cut away the peel and white pith in strips, working from top to bottom and following the natural curve of the fruit. Hold the fruit over a bowl to catch the juices and cut down along the membranes on either side of each section to release it, letting the sections drop into the bowl.
Bring 6 quarts of water to a boil in a large pot, and add 3 tablespoons kosher salt.
Meanwhile, combine the lemon zest, juice, olive oil and butter in another pot and heat over medium-low heat until the butter melts. Remove from the heat.
Drop the pasta into the boiling water and cook until just al dente. Drain the pasta, reserving about 1/2 cup of the pasta water.