“Yule” derives from the Old English Gēol which now refers to Christmas but was originally a pagan winter solstice celebration. The Yule Log Cookie is descended from the Yule Log Cake where it is known as a Bûche de Noël in France and a Tronchetto di Natale in Italy.
By the way, no need to worry about the beaver losing out since both humans and beavers have been harvesting the original version of a Yule Log in forests for centuries. The humans’ goal was to have a log large enough to burn for all the 12 days of Christmas. (Obviously not one of the beavers’ concerns!)