In a 4-quart Dutch oven, combine chicken, 3¾ cups water, bouillon granules, salt, caraway seeds and pepper. Bring to a boil; reduce heat. Cover and simmer for 40 minutes. Stir in green beans, carrots, celery, mushrooms and onions. Return to boiling; reduce heat. Cover and simmer about 10 minutes or until chicken is tender.
Remove chicken pieces from the stew; set aside to cool slightly. When cool enough to handle, remove meat from bones; discard bones. Cut up the meat; add to vegetable mixture in pot. Combine ¼ cup cold water and flour in a small bowl; whisk until smooth. Add to stew. Cook and stir until thickened and bubbly. Cook and stir for 1 minute more.