You're totally grossed out by these, and the words "umbilical cord," "sperm," or "embryo" may have crossed your mind a few times upon encountering these thick, ropey strands of egg white that are often attached to an egg yolk. Relax. The chalazae's only role is to anchor the egg yolk to the center of the egg. Remember, the eggs we eat are not fertilized, and chickens are not mammals, therefore no umbilical cord, sperm or embryos, my friends.
There's even a useful aspect to the chalazae: the larger it is, the fresher the egg. Once cooked, they disappear and have no effect on recipes, with the exception of custards, which usually call for straining to remove them in order to achieve a perfectly smooth texture.