How do I ask for eggs, sunny-side-up, but with the top cooked more thoroughly? If I ask for eggs, sunny-side-up, they will have yellow barely cooked yolks.  At home I avoid this by placing a lid and creating hot air or steam that cooks the top more thoroughly.  How do I ask for this style at a restaurant?  What term do I use?
You can ask for eggs over easy.
This is when the cook flips the egg over once it's ready and allows the wolk to cook a bit on the hot surface of the pan.
The phrase is an American one, if I'm not mistaken, but is probably understood anywhere.