Beat the coffee grounds into an egg while bringing water to a boil, then get the whole mass into the water. Can boil for an extended time (say 3-5 minutes) and use old stale coffee and it will improve the flavor and keep the grounds suspended in the boiled egg making the pour off easy.

That's a very common misconception. The egg shells are for their calcium carbonate, in order to cut down acidity and improve the flavor. The classic trick to causing the grounds to settle quickly is simply to add a little bit of cold water at the end of brewing.

The egg whites that remain on the shells also acts as a flocculant for suspended solids.

The trick for that is to use two identical bowls, one as a mortar and the other as a pestal.