Pierce the wider end of a cold large egg with something sharp (like a sewing needle). Push it in deeply to pierce the yolk. Place the egg in a 2-cup glass measuring cup. Cover with cold water by one inch.
Microwave on 50% power for 4 minutes.
Let the egg rest in in the hot water for 2 minutes (it continues gently cooking and prevents too much pressure from building up).
Microwave again on 50% power for 2 more minutes.
Transfer the egg to an ice bath to cool and stop the cooking. Tap egg on hard surface and peel.
Video
Notes
SAFETY NOTE: Microwaving eggs can cause pressure to build up inside them and there's a chance they can explode.
Egg will be less likely to explode if the yolk is pierced and the egg is fully covered by water for the whole time it is cooking. It is not a guarantee, however, and sometimes one will still go boom. Use caution and try this method at your own risk.
Cover the Dish with Plastic Wrap: for extra insurance against messy egg explosions in the microwave.
RECIPE NOTES:
Microwave Variances: All microwaves are different. Adjust cooking times based on the wattage of your microwave and the number of eggs you're cooking.
Container Choice: Use a microwave-safe container like a glass measuring cup and ensure the egg is completely submerged in water.
I tested this method using a 2-cup glass measuring cup and 1 1/2 cups of water, which covered my egg(s) by an inch, as directed.
If you use a different size or shape of dish, you may need to adjust cooking times slightly.
See notes in post on how to adjust the method to microwave boil 2 eggs at once.