First problem: how to prevent shell from breaking (as in this photo) and white spilling out into the water while cooking? Egg shells break during cooking for two reasons: 1) Temperature difference, if they go straight from the fridge to boiling water, the shell always (or nearly always) breaks. Solution: remove your eggs from the fridge at least one hour before cooking. Note : Eggs don't need to be kept in the fridge.
2) Because in the bottom part of the egg, the broader end, there is an air pocket (in red on the diagram), and when cooking this air expands and eventually breaks the shell. You will notice that the older the egg is, the bigger the air pocket, so the risk of the shell breaking is greater with an old egg than a fresh one.
Solution: You need to pierce the air pocket so that the air can escape during cooking. To do this you can use big needle or a small nail (as in the photo), push in it a few millimetres.
The aim is to just pierce the air pocket and to make a very small hole.
Boil a pan of water, and lower in your pierced eggs gently, using a tablespoon. Leave to cook 10 minutes, no more.
If eggs cook longer, yolks will became yellow-green (as in this photo), and develop a sulphurous taste and smell, and white will become rubbery.
To demonstrate the effect of piercing: on left a pierced egg (shell intact), on right an un-pierced egg (shell broken).
Cool eggs immediately by putting them into cold water.
To remove the shell, tap the bottom of the egg on your work surface to break it.
Do the same with the top of the egg.
Roll the egg round with your hand to break all the shell.
Then remove the shell, as if peeling the egg.
Wash the egg to remove any small pieces of shell, and leave in cold water, or covered in the fridge, until using.
Another way to cook, probably priceless: Fill the pan with cold water to submerge the eggs (drilled) and bring to boil.
When boiling is reached, cover, remove from heat and let stay for 10 minutes.
It seems to me that with this way of cooking, for mysterious reasons, the eggs are a bit difficult to peel.