chewing the fat....Takeourword.com Issue 112 p.2 says phrase began as chew the rag, then chew the rug, finally chew the fat about 1885.
I remembered having read about Inuit women having to chew remnants of fat of skins before tanning them.
Instead of the boat trip, some of the Inuit women set up a large tent, and we all gathered inside for a demonstration of how to prepare a sealskin for drying and then for use. First she used an ulu, a crescent-shaped knife, all purpose tool, to shave off all the remaining fat and flesh from the back of the skin. Then she washed it and began the softening process, which ultimately includes long and careful chewing of the skin. All of this she demonstrated. The skin was then stretched on a frame for drying.
After several days of drying and more washing, the softening by chewing process is one of the final steps before the skin is used for boots, mittens, or whatever. She also showed us the strip of sinew that the threads were pulled from. Very fine, strong threads can be separated from it to be used for sewing the fur