Bring a large pan of lightly salted water to the boil. Cook the potatoes for 10 minutes until beginning to soften. Drain and set aside to cool.
Heat the butter in a small frying pan and add the salmon. Cover with a tight-fitting lid and reduce the heat to very low. Cook for 8–10 minutes until the salmon is just cooked. Remove from the heat and set aside until the salmon is cool enough to handle. Flake the salmon and place in a bowl with the pan juices.
Grate the cooled potatoes, add to the bowl with the salmon and toss together to mix. Divide the mixture into 6 and form into flattened patty shapes. Heat the oil in a large nonstick frying pan and cook the rösti over a moderate heat for 2–3 minutes on each side until golden and cooked through, turning with a fish slice.
Meanwhile, place the crème fraîche in a bowl with the spring onions and chives and mix well. Drain the rösti on kitchen paper, then spoon over the crème fraîche and serve with lemon wedges.