Sprinkle the chicken pieces with salt and pepper. Heat the butter in a large, heavy skillet large enough to hold the chicken pieces in one layer.Add the chicken and cook, turning occasionally, without browning, about 5 min.
Add the onion,garlic and tarragon and stir well. Sprinkle with the flour, stirring to distribute the flour evenly. Add the water, carrot and mushrooms. Cover and cook about 15-20 min.
Remove the chicken pieces, carrot and mushrooms to a warmed serving dish and keep hot.
Let the sauce cook down, stirring often, about 5 min.
Add the cream and salt and pepper to taste. Simmer about 3 min. Pour a little sauce over the chicken and serve the rest separately.