The use of the past perfect is very weird. It also should be "her" plate, not "he" plate.
A egg that is "sunny side up" is a fried egg in which the yolk is not broken. It's also not cooked all the way through. As soon as you pierce the yolk with your fork, it runs all over the place. (I personally find this gross. I ask for my eggs to be cooked firm, not runny.)
The girl doesn't want the yellow yolk to run over her nice egg whites.