You guys know making poached eggs is very easy with fresh eggs? The firmness of a fresh egg keeps the egg together! I just tried it out the other day and I must say, the fresher the egg, the more it held together.
So what I did was had about 2 to 3 inch high water in a low simmer, not even bubbling, barely boiling. Put 1/4 cup vinegar in, maybe more or less depending on how big your pot was, and salt. Crack the egg in a bowl and slide it into the water. Let the egg harden enough so it won't break before using a ladle to move it and then let it cook to the right temperature/hardness you want. Then take it out and serve.
The fresher the egg, the less egg white will swirl around the water because it is firmly attached to the yolk. Big difference between just laid egg vs refridgerated 5 day egg.