Heat the butter and 1 tablespoon of the oil in a saucepan. Add the shallots and cook for 5 minutes until softened. Add the potato, stock and herbs and simmer for 10 minutes until the potato is just cooked through. Remove the herbs and add the cream and peas. Cook for 3 minutes until just tender.
Meanwhile, heat the remaining oil in a large, nonstick frying pan. Add the prosciutto and cook for 1–2 minutes on each side until golden. Drain on kitchen paper. Season the sole fillets, dust with flour and cook for 3–5 minutes on each side until just cooked through.
Scoop out a handful of peas from the pan and reserve. Whizz the remaining contents in a food processor until really smooth, then return with the whole peas to the pan. Season well and add lemon juice to taste. Spoon on to serving places, then top with the fish and prosciutto to serve.