Render chicken fat in a heavy casserole until most of the fat is melted. In 1tablespoon of the fat slowly wilt onion until limp, about 15 minutes.
Dry chicken pieces. Set aside breasts and liver, and cook the rest of the pieces together with onion in a covered pan for 5 minutes. Don’t let the onion burn.
Add salt, pepper, sugar, knob celery, carrots, parsley and the chicken broth or water. Cook covered over very low heat for 25 minutes.
Add breasts and liver and cook for another 10 minutes.
In a separate bowl mix flour with 3tablespoons cold water until mixture is smooth. Take a ladleful of liquid from the pot, whip it well with flour-water mixture, and finally pour entire mixture into the pot. Simmer over very low heat for 5 to 10 more minutes.