Boil the potatoes in their skins in lightly salted water until they are tender but not breaking up. Drain, cool and then cut into chunks, keeping the skins on. Put to one side for now.
Heat 2 tablespoons of the oil in a 26 cm(10½ inch) non-stick frying pan over medium heat and fry the sausage meat until it is quite deeply golden, breaking up clusters with a wooden spoon. Lift out into a bowl with a slotted spoon. Add the onion to the pan and sauté until golden, then add the carrot. Sauté for 5 minutes or so, until both become soft and a bit gooey. Add another tablespoon of oil and the mushrooms and sauté until the mushrooms have given up their water and turned golden. Season lightly if needed. Spoon the contents of the pan into the bowl with the sausage.
Preheat the grill (broiler) to medium–high.
Add the last of the oil to the pan and sauté the potatoes until they have a golden crust in some places. Stir in the parsley and the contents of the sausage bowl and cook for a minute or two. Reduce the heat to medium–low and pour in the eggs, shifting everything around with a wooden spoon so that the eggs leak under here and there. Once the bottom looks softly set, put the pan under the grill so that the frittata sets just enough to be cut into slices. Don’t overdo it though, as this is nice when it is still a little soft. Serve warm with your favourite bread.