Put all ingredients except the vegetable oil in a bowl and mix well with a whisk (mix until the egg whites are blended well, but do not to create bubbles).
Strain the mixture with a sieve, and then strain again using a finer tea strainer. ※Mixing the mixture with a spoon while straining will speed up this process.
Thinly coat the frying pan with oil using a paper towel dipped in oil, and heat on medium heat.
When the frying pan is heated, pour in 1/3 of the egg mixture.
Gently scramble the egg when it starts to bubble, then roll it up when the egg is soft set. Remove the frying pan from the heat while rolling. Use a spatula to roll if it is difficult.
Using the same paper towel you used to coat the pan with oil, remove the excess egg from the pan while coating the pan with a new layer of oil. Pour half of the remaining egg mixture so that it flows beneath the already rolled egg.
Repeat the rolling process as before, add the remaining egg mixture and rolling while cooking (The egg should be rolled 3 times in total).
The egg should be soft-set at this point.
Reduce the heat to low (to prevent burning) and cook the egg through while gently pressing down on the egg with a spatula. Then it's ready.
Addendum: Pressing the tamagoyaki to the side of the frying pan with the spatula will make a thick tamagoyaki without air pockets.
Addendum: Even if you have trouble rolling the eggs nicely and they become scrambled, just finish by pressing down on them and fry, then it should be be fine.
Addendum: If you turn off the fire, cover the pan with a lid and let the tamagoyaki steam cook, you can prevent the tamagoyaki from being undercooked.
Addendum: Straining the egg mixture will make the tamagoyaki smoother, but it should also be fine mixed with a whisk without straining. Don't bother straining the mixture if you don't have time.
The flavors will be more pronounced if you chill the tamagoyaki in the refrigerator after it cools down slightly. I recommend serving it chilled to a freshly made one.
Add bonito soup stock and a little salt and sugar instead of water when making a dashi-maki. Or you could use a stock made from 1/2 teaspoon powdered soup stock to 50 ml water.
Addendum: When making atsuyaki-tamago (thick tamagoyaki) with a doubled amount of ingredients, it will be easier to roll by repeating the pouring and rolling process several more times.
Addendum: If it burns easily, reduce the heat to low heat.
Story Behind this Recipe
I wanted to make a tamagoyaki for bento that would be delicious and springy even cold. I wanted it to be bouncy, so I based my method on chawanmushi (steamed savory custard) instructions.
Straining the egg mixture will produce a smooth texture. Adjust the heat by removing the frying pan from the heat while cooking. It is difficult to roll the egg when it is soft set, but this will prevent any brown layers from appearing when the egg has been wrapped up. Addendum: adjust the amount of seasoning when using large-sized eggs.