Peel and boil the potatoes. Wash and dry the fish on kitchen paper.
Melt the butter in a frying pan with the oil (the oil stops the butter going too brown), add the fish and fry until tender (about 5-10 minutes), spooning the melted butter over it as it cooks. (The thicker the fish, the longer it will take to cook.).
Lift the fish carefully onto a warm plate, add a little chopped parsley (if used) to the butter in the pan and heat thoroughly. Pour the buttery juice over the fish and garnish with the lemon slice. Drain the potatoes and serve.