Make the push pin ready. You can use a thumb tack, too, but it's harder to grip.
Pierce the rounded end of the egg with the pin while turning it round and round.
Pierce it completely, then remove the pin.
It looks like this. The inside of the egg won't come out.
Put the eggs in boiling water. Not cold water, boiling hot water. This is the key.
Here's an 8-minute egg. It's a bit softer than soft-boiled.
A 9-minute egg. It's perfectly soft-boiled.
A large egg boiled for 10 minutes.
Boiled for 12 minutes.
Can you see the differences between boiling for 8, 9, 10 and 12 minutes?
These 10-minute eggs were all successfully peeled.
There's no need to put the eggs in cold water.
Sometimes the shell cracks when the eggs are boiling! (And sometimes the white leaks a bit from the hole.)
If you are not going to peel the eggs immediately, I recommend putting them in cold water.
Continued - if you leave the eggs as-is without putting them in cold water, the hot water inside the egg may evaporate and the membrane may get stuck.
If you leave the eggs without putting them in cold water, the yolk may continue cooking in residual heat. So if you want soft-boiled eggs, I recommend putting them in cold water.
Story Behind this Recipe
I've been making perfectly smooth and flawless boiled eggs like this for many years.
By piercing the eggs with a pin, you are making a hole in the membrane of the egg. When an egg is boiled like this, hot water gets in between the membrane and the shell, and so the egg becomes easy to peel. If you put the eggs in cold water, the white may come running out of the hole, so put the eggs into boiling water.