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For the caramelised red onions, heat the butter and olive oil in a frying pan over a medium-low heat. When the butter has melted, reduce the heat to low and cook the onion with a pinch of salt for 20-30 mins, stirring occasionally until soft and golden. Add a splash of water if the onion starts to catch. Stir in the sugar and cook for a further 5 mins until the sugar has dissolved and the onions are sticky. Set aside.
Put the feta, soft cheese and lemon juice in a bowl, season well and beat together until smooth. Gently fold through the harissa to create a ripple effect, then chill until ready to serve.
Tip the grated potatoes into a clean tea towel and squeeze out all the excess water. Transfer to a bowl with the parsnips, shallots and melted butter, and stir to combine. At this point, you can add the egg if the mixture isn’t binding together. Season. Form into four patties, pressing together tightly.
Heat half the oil in a non-stick frying pan over a medium heat and, when hot, fry two rösti for 4-5 mins, pressing down with a fish slice or spatula. Carefully flip and cook for a further 4-5 mins until both sides are crisp and golden. Keep warm in a low oven while you cook the remaining two rösti in the same way.
Serve the rösti topped with the feta and harissa mixture, caramelised red onions and parsley scattered over.