To clean the artichokes, cut off part of the stalk leaving just 4 or 5 cm; remove the outer, tougher petals; cut off the tip, the external part all the way round the base below the petals, and the outside of the stalk. As you clean them, put them in a bowl of water with the juice of the lemon (also adding the two halves of the squeezed fruit). Then strain them, pat them dry and cut them into slices. Sauté the garlic and chilli pepper with the olive oil in a heavy-bottomed frying pan. Add the sliced artichokes, season with salt and pepper and sauté for a few minutes. Add the anchovy filets and break them up with the tip of a wooden spoon. Continue cooking until the pieces of artichoke are tender, but not too soft (they mustn’t fall apart), stirring frequently. Cook the spaghetti in plenty of salted water (be careful because both the anchovies and the pecorino are flavoursome) and strain them when “al dente”, setting aside some of the cooking water. Toss the pasta in the frying pan with the artichokes, add 1 or 2 ladles of the cooking water and the grated pecorino, give everything a good stir and serve the spaghetti straight away, piping hot!