"Frying" an egg in soy souce?
As the comments note, this hasn't been done before because by definition, it's not frying.  Frying always involves cooking with a liquid fat.  This is a bit like asking what happens if you boil something in bacon grease.  You can do something similar, but technically it's not quite the same thing.

Terminology aside, you will probably want to use only a very small amount of soy sauce - too much, and you'll wind up tasting pure salt.  I think you might have trouble getting this evenly distributed with straight soy sauce, so consider diluting with 1-2 parts water as well before you add to the pan (which would make this a good candidate for Jolene's method).  You'll also probably still want to have some fat in the pan for the flavor it will add and to help reduce sticking.