This thin part is what produces those wispy bits of white when you poach eggs. No amount of vinegar, no rapidly spinning whirlpool of water will persuade them to coagulate into a neat shape.
At best, you will have to trim them once they are cooked. But, if you take a moment to drop the egg into a strainer and allow the watery white to drain away, you will get a perfect egg every time.
After poaching for two and a half to three minutes, they are ready to serve or to be placed in a bowl of iced water to be used later. Also see Eggs Benedict recipe and video.
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