Melt the oil and butter together in a large saucepan, and fry the onion and garlic over a very gentle heat until soft but not brown, about 15 minutes. Add about three-quarters of the mint, the peas and the stock. Cook, covered, until the peas are tender, about 5 minutes.
Put half the mixture into a food processor and pulse until you have a coarse purée. Return to the pan and season. Add