1. In a shallow bowl, combine the flour with 1 teaspoon each of salt and pepper. Dredge the chicken in the flour; shake off the excess.
2. In a medium enameled cast-iron casserole, heat 2 tablespoons of the oil. Add half of the chicken and cook over moderately high heat until golden, about 2 minutes per side. Transfer the chicken to a bowl. Brown the remaining chicken in 2 more tablespoons of the oil.
3. Pour off the fat. Add the remaining 1 tablespoon of oil and the carrots to the casserole and cook over moderate heat for 3 minutes. Add the celery, leek and garlic and cook until starting to soften, 2 minutes. Add a large pinch each of salt and pepper and the barley. Cook over moderately high heat, stirring, until the barley starts to toast, 1 minute. Add the stock and water and bring to a boil. Cover and simmer over low heat, stirring occasionally, until the barley is almost tender, about 20 minutes.
4. Return the chicken to the stew; cover and simmer until cooked through and the stew has thickened, about 10 minutes. Stir in the dill and lemon juice and season with salt and pepper. Spoon the stew into bowls, sprinkle with the tarragon and cheese and serve.
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