Put the chorizo and oil in a frying pan over a very low heat and cook, turning occasionally, until crisp. The melted fat released by the chorizo will be an essential part of your sauce.
Cook the pasta in a large saucepan of salted boiling water according to the packet instructions until it is al dente.
Meanwhile, crack the eggs into a bowl, add the Parmesan and season with salt and a generous grinding of pepper. Mix together with a fork.
Just before the pasta is ready, increase the heat under the frying pan so that the oil and melted chorizo fat start to sizzle. Drain the pasta thoroughly, return to the pan and immediately stir in the egg mixture and the sizzling-hot contents of the frying pan. Stir vigorously so that the eggs cook evenly. Serve immediately with a scattering of grated Parmesan.