Sprinkle the chicken pieces with salt and pepper. If desired, set the chicken liver aside to prepare rice with liver as a side dish for the chicken.
Heat the oil in a skillet and add the chicken pieces, skin side down. Cook over moderately high heat 3 to 5 minutes or until golden brown. Turn the pieces and continue cooking about 3 minutes or until golden brown on the second side.
Add the mushrooms and continue cooking. Add the bay leaf, thyme, onion, and garlic. Cook, shaking the skillet and stirring the chicken pieces so that the other ingredients are evenly distributed around and between the pieces.
When the chicken has cooked about 10 or 12 minutes (total cooking time, start to finish), remove the pieces to a serving dish.
Sprinkle the remaining ingredients with flour, stirring to blend evenly.
6. Add the wine and tomatoes and broth. Stir and bring to the boil.
7. Return the chicken pieces to the skillet. Add the juice that may have accumulated around them. Add anchovies, salt and pepper to taste. Cover closely and simmer 5 minutes.
8. Add the olives and continue to cook 5 minutes.