Combine livers, 1/2 cup onion, garlic, salt, and water to cover in a medium saucepan; bring to a boil. Cover; reduce heat, and simmer 20 minutes or until livers are tender. Strain broth, reserving livers. Pat livers dry; process in food processor or put through a food mill until coarsely ground. Sauté remaining onion, green pepper, and celery in butter in a large skillet. Stir in thyme, ground livers, and rice. Gently spoon mixture into a greased 2-quart baking dish; bake, uncovered, at 350° for 30 minutes or until lightly browned.