Preheat grill to medium-high. Put prosciutto on an oven tray and grill for 2-3 minutes on each side until crisp. Set aside to cool, then break into pieces.
Heat oil and butter in a frying pan over a medium heat until butter melts. Add onion and cook, stirring occasionally, for 15-20 minutes or until onion is golden. Set aside to cool.
Meanwhile, cut potatoes into 4cm pieces and put in a large saucepan over a high heat. Cover with cold water and bring to the boil. Simmer for 8-10 minutes or until potato is tender, then drain and rinse under cold running water. Transfer to a large bowl and set aside to cool.
Put aioli, mustard and ²/³ of the parsley in a small bowl. Season with salt and pepper and whisk to combine. Add to cooled potato mixture and fold in onion and ²/³ of the prosciutto. Transfer to a large serving dish. Sprinkle with remaining prosciutto and parsley to serve.