Heat the oil in a large flameproof casserole dish or saucepan over a moderate heat. Add the chicken pieces and brown them on all sides for 8–10 minutes. Remove and set aside. Reduce the heat to low, melt the butter, then add the carrots and the whites of the leeks. Stir well, cover and cook gently for 10 minutes. Season with salt and pepper.
Put the chicken pieces on top of the vegetables. Scatter the parsley sprigs and sliced green leeks over the top. Pour in the stock, bring to the boil, then reduce the heat, cover and simmer for 1 hour, or until the chicken is tender.
Discard the parsley. Lift out the vegetables and chicken using a slotted spoon and put on a serving dish.
Boil the stock left in the pan over a high heat for 5 minutes until reduced by a third. Reduce the heat.
Mix the egg yolks with the crème fraîche or sour cream. Gradually blend this into the reduced stock and simmer, stirring for 2 minutes, until thickened. Do not let it boil. Pour over the chicken and sprinkle with chopped parsley.