As far as I know it's long term boiling that makes them greenish/greyish. Having said that, no two eggs are created equal --- they might have the same basic chemical makeup but as we all know eggs from the shops and eggs from the home are different in every way, so it stands to reason that they'd cook differently too. At least to me it does.
'Oil' could mean a lot of different things. What oil are you speaking of?
Also you'd need to try minus the cold water to determine what is helping or not.
Okay let me clarify... I use a bit of olive oil because that is what I cook with. But veggie oil, or canola oil or just about any other kind of cooking oil would work also.
"Always" is a strong word. I USUALLY run them under cold water right away. There have been times I didn't for different reasons. I have not seen a difference in the ease of peeling them by running them under cold water, or not. However when I use the oil it works no matter if I run them under cold water right away or not.
Better?

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