Pour a splash of extra virgin olive oil in a thick-bottomed pan: when it is hot, add artichokes and brown them on all sides, stirring often. Meanwhile, whisk the eggs with salt and pepper and a generous grating of Roman pecorino cheese. Pour the egg mixture over artichokes, add a few leaves of basil and let the omelet get firm on one side. Then, with the help of a dish, turn it upside down and let it get firm on the other side, too. Serve hot, so that the omelet will remain soft and it will exalt the contrast with the crisp wedges of artichokes.