Melt the butter and olive oil together in a Dutch oven or other heavy large skillet on medium heat. Add the chopped onion and crushed garlic and cook for five minutes stirring occasionally. Add the chicken and brown on all sides - about five minutes. Stir in salt and pepper.
Sprinkle the top of the chicken with the flour and stir for 1 minute, thoroughly coating the chicken with the flour. Stir in the water and wine and then the prepared mushrooms, carrots and tarragon. Bring to a low boil, cover and simmer on medium low heat for 15 minutes or just until the vegetables are cooked as you like them.
Turn off the heat and stir in the crème fraîche. Serve hot.
Makes 6 servings.