Remove the eggs from the fridge. In a small pot, bring water to a boil over high heat. How much water you use will depend on your saucepan. It should be enough to fully cover the eggs.
Lower the heat to medium-low to maintain a gentle boil. Using a slotted spoon, gently lower the eggs into the water.
Gently boil the eggs, uncovered, for exactly 5 minutes. Use a slotted spoon to remove the eggs from the water gently but quickly. Place them in egg cups.
When cool enough to handle, slice the top of each egg with a sharp knife.
Perfection achieved! A cooked white and a thick, runny yolk! Sprinkle the eggs with salt and pepper and dig in.
Video
Notes
The sodium info is for the eggs only; add salt to taste.
In contrast to hard-boiled eggs, where you should use fairly old eggs (this makes them easier to peel), here I recommend using eggs that are as fresh as possible.
It's best to use eggs that are not refrigerator-cold to prevent the shells from cracking, especially since we're putting them into (gently) boiling water. But some eggs seem to have thin shells and are prone to cracking, no matter what you do.
You can keep the leftovers refrigerated, in their shell, and in an airtight container for up to two days. Reheat them by briefly placing them in simmering water for no longer than one minute.