In a bowl, gently whisk eggs. Add dashi, soy sauce, mirin, sugar, and salt. Mix well, but try to avoid creating too much foam.
Heat a rectangular tamagoyaki pan (or a small non-stick pan) over medium-low heat. Lightly oil the pan with vegetable oil.
Pour about 1/4 of the egg mixture into the hot pan, tilting to coat the bottom evenly. Cook until just set, but still slightly wet on top.
Once the egg is mostly set, gently roll it towards one end of the pan using chopsticks or a spatula.
Lightly oil the now-empty part of the pan. Pour another 1/4 of the egg mixture into the pan, lifting the cooked egg roll slightly to allow the new mixture to flow underneath.
Once the new layer is mostly set, roll the already cooked egg over this new layer, continuing to form a thicker roll.
Repeat steps 5 and 6 with the remaining egg mixture until all is used and you have a thick omelet roll.
Remove the omelet from the pan. Shape it gently with a bamboo mat (makisu) if desired. Let it cool slightly, then slice into bite-sized pieces and serve.
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