To stop your egg bursting open, even straight from the fridge in boiling water, drop it in on a spoon and keep on turning it/them over and over with the spoon. This distributes the air evenly around the egg and lets the outer skin set quickly all around.
I had to learn all this because I have kept chickens for years, and initially had children who were very picky as to how they would eat their eggs. When you have cooked eggs for breakfasts pretty much every day for 30 years it soon sinks in.
If you use a draining spoon (yes, I have used a spaghetti spoon too!) and it doesn't look quite cooked and 'watery', it's very easy to pop it back in again for a few seconds.