In a tall stock pot heat the vegetable oil. Add the chicken and saute over moderate heat for about 5 minutes or until browned. Remove and set aside. Add the onions and saute until golden. Return chicken to stockpot with 1/4 cup of water. Cover and simmer over very low heat for 20 minutes or until chicken has given off its liquid. Add carrot, celery, remaining water and bouquet garni. Slowly bring water to a boil, skimming on occasion and simmer for 30 minutes longer.
Strain, discard the solids and cool to room temperature. Refrigerate and remove fat the following day. Reheat to adjust the seasoning.