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In a large heavy saucepan, heat the butter and oil. Sauté the shallots and garlic with some seasoning for 5 mins, then stir in the chicken and cook for 2 mins.
Add the barley and cook for 1 min. Pour in the wine and stir until it is absorbed. Add the mushrooms and thyme, then pour over ¾ of the stock. Cook for 40 mins on a low simmer until the barley is tender, stirring occasionally and topping up with remaining stock if it looks dry. Remove from the heat and stir in the grated Parmesan. Serve immediately, with chives and Parmesan shavings scattered over, if you like.