‘The Oxford English Dictionary’s earliest citation for “Chew the fat” is from 1885 in a book by J Brunlees Patterson called Life in the Ranks of the British Army in India. He implied it was a kind of general grumbling and bending of the ears of junior officers to stave off boredom, a typical part of army life.[8] Patterson also uses “chew the rag” in the same sentence he used “chew the fat”, but it is not the oldest occurrence of that phrase.
Prior to the adoption of metallic cartridges, most ammunition was composed of powder and a ball wrapped in paper or cloth soaked in animal fat, which was bitten open during musket drill. Soldiers were known to chew on these ends to pass the time and reduce nerves, and in some cases to stave off cravings for chewing tobacco.
Though long-since replaced by 1885, the idea of biting or chewing on fat-soaked rag ends may well have entered military parlance in this fashion prior to Patterson’s recording.’