Schoolgirl, the white thingy in an egg is called "Chalazae'. These two whitish cords center the yolk between the two ends.
For custards and sauces, they should be strained out so they don't mar the finish of the dish. other wise, who cares.
Another little observation is that the more prominent the chalazae, the fresher the egg.
I got this from a Cooks Illustrated article MAR/APR 2003 pg 16 'Eggs101'. I did remember it because my sister went to great lengths to try to convince me it was the part from the Rooster. I copied the article and sent it to her. Never heard back on the subject. Ain't sibling rivalry fun!
May you eat well,