Use a large saucepan to heat the oil over a medium-high heat. Add the giblets and the neck and permit them to brown, stirring occasionally, for about five minutes.
Add the onions, celery, carrots, salt, pepper, bay leaves, and the dried thyme. Cook this mixture, stirring often, for three to four minutes. Add the flour and stir to combine. Cook, stirring, for about five minutes.
Add the Tawny Port, bring the mixture to a boil, and cook for about five minutes, stirring from time to time.
Add the water and bring the mixture to a boil. Reduce the heat to medium and simmer for about one hour.
Remove from heat and strain through a sieve. Keep warm until ready to serve.
Makes about two cups of gravy.