Take home message, it is no big deal, pick it out with your fork and then eat the now clean egg. If you are still grossed out over it you can cook up the entire thing and feed it back to the chickens.
It is pretty rare, but does happen.
I think that pullets are most likely to "mess up" with their eggs.
It looks like a bit of tissue got shed at some point during the egg formation process and got encased with the egg contents. That yolk is much too pale for my liking. Really pale, or white yolks can be caused by an infestation of capillary worms. If you haven't dewormed your flock recently it may be a good idea to do so. What are you feeding your hen? Is she on a good quality layer feed?