In a medium bowl, combine 2 tbsp of water, sugar, mirin, salt, and ¼ tsp of rice vinegar. Mix until the sugar dissolves. Add eggs, and whip.  

Using a Tamagoyaki pan, heat at medium temperature. 

Pour a thin layer of the egg mixture into the pan. When the egg is half cooked, gently roll into a log, pushing it to the farthest side of the pan from you, leaving it in the pan.

Add oil if needed. Then, pour another thin layer of egg mixture on the empty space of the pan. Once this layer of egg is half cooked, roll the previous egg log into the new mixture, forming a thicker log. Push to the farthest side from the pan from you.

Repeat for a third time or until egg mixture finishes. Remove from the pan and cling wrap to shape. Set aside and let cool.

The egg should be only half-cooked before you try rolling it up in the pan. If it is fully cooked, it will not stick together when you roll it up.

A rectangular Tamagoyaki pan is best, but a round pan can work as well.