Cooking eggs in boiling water, without the shell.

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Nathan you just cooked a basic poached egg! Yes, safe to eat, try them on a piece of toast of English Muffin. A better strategy would be to cook the eggs in simmering, not boiling water. Typically you want the whites somewhat solid, but not rubbery like a hard cooked egg and the yolks runny. If you want the whites a little more solid than that, cook a bit longer, say 10-15 seconds. Some cooks add a tablespoon of white vinegar to help the whites keep their shape.

Yes, safe to eat, unless you have health issues that advises you against eating raw egg yolk. You've essentially poached an egg in boiling water. If you cook it longer, the whites will get tough and rubbery which is not pleasant.