Egg Farming Joe is a hypothetical version of Joe that exists in Medieval Europe. He has all the eggs.
On the first episode of the show, which is now deleted, Lars asked Joe what he would want his career to be in Medieval Europe, and one of Joe's answers was that he would farm eggs and have a business doing that because it seemed easy and lucrative. Lars coined this hypothetical version of Joe as "Egg Farming Joe" which Joe agreed rolled off the tongue.
Very occasionally, Egg Farming Joe will make an appearance on later episodes, either outright or as a subtle reference.