1. Wash the gizzards and hearts very well and put them in a saucepan with the broth, onion, bay leaf, and salt and pepper to taste. Bring to a boil, cover, and simmer until tender, about 45 minutes.
2. Remove the gizzards and hearts, chop finely and reserve. Reserve the broth.
3. Coat the liver pieces with the flour seasoned with salt and pepper. (Reserve any remaining seasoned flour.) Heat the butter and 2 tablespoons of the oil in a heavy skillet.
4. Fry the livers quickly in the skillet until browned on all sides. Remove and reserve. Add the remaining oil, the garlic, and mushrooms to the skillet and cook for 2 minutes. Add the celery and carrots and cook for 3 minutes, stirring occasionally.
5. Add the reserved broth and chopped gizzards and hearts, bring to a boil, cover, and simmer until vegetables are barely tender, about 12 minutes.
6. Add reserved liver pieces, any remaining seasoned flour, the basil, marjoram, thyme, and salt and pepper to taste.
7. Heat, stirring, until the mixture thickens slightly. Cook for 3 minutes longer.