Melt half the butter with the oil in a large frying pan with a tight-fitting lid. Add the onions and season with salt and pepper, then cover and cook over a very low heat, stirring occasionally, for 35–40 minutes until very soft and golden. Remove to a bowl and increase the heat under the pan to high.
Season the liver with salt and pepper and melt the remaining butter in the pan. Once the butter starts foaming, add the liver and cook for 1–2 minutes until browned. Turn over and return the onions to the pan. Cook for a further minute, then serve immediately with the parsley scattered over.