Beat the eggs in a bowl. Make sure not to over-beat the eggs.
In another bowl, whisk together water, rice vinegar, sugar and mirin until sugar is dissolved.
Combine the egg mixture with the seasoning mixture. Whisk gently. Strain the egg mixture through a sieve. Pour the mixture into a measuring cup or a jar with a spout and handle for easy pouring during cooking.
Heat a rectangular tamagoyaki pan (you can also use a round pan, see notes*) over medium. Then dip a folded paper towel in oil, and apply to the pan.
Once the oil is hot, pour a thin layer of the seasoned egg mixture into the pan. Tilt the pan to allow the liquid to flow to the edge of the pan.
After the egg has set a little bit but still soft on the surface, start rolling it into a log shape. Make sure the egg is not cooked too much, otherwise, it won’t stick as you roll the log. (It’s fine if your eggs are not neatly folded).
Move the rolled omelette to one side, and apply more oil to the pan with the paper towel. (Remember to apply the oil under the omelette.)
Pour another layer of egg mixture cover the bottom of the pan. Lift the omelette to let the mixture run underneath.
When this new layer has set slightly and still soft on the top, start rolling the log back onto the set egg to the other end of the pan.
Repeat this process until you’ve used all the egg mixture.
Remove from the pan and place the tamagoyaki on a sushi mat. Wrap it up and shape the tamagoyaki when it’s still hot. Let it rest for about 5 minutes.
Slice the tamagoyaki into 12 pieces, and set aside.