How to Make It
In a small saucepan, bring the water to a boil. Add the bonito shavings and simmer over low heat for 1 minute. Cover, remove the saucepan from the heat and let stand for 10 minutes. Strain the dashi through cheesecloth and let it cool to warm.
In a bowl, whisk 125 milliliters (1/2 cup) of the warm dashi with the sugar, mirin, salt and 1 teaspoon of soy sauce. Whisk in the eggs.
Heat a large nonstick skillet over moderate heat. Rub the pan with an oil-soaked paper towel. Add a drop of the egg mixture to the pan and, when it sizzles, pour in one-quarter of the egg mixture so it forms a thin layer. As the egg begins to cook, use chopsticks or a spatula to roll the omelet away from you, popping air bubbles as you go, to form the omelet into a flattened log. Dab more oil in the skillet and add another quarter of the egg mixture, lifting the rolled omelet and tilting the pan to allow the egg mixture to seep under it. As the layer of egg cooks, roll the omelet with the new layer of egg toward you. Push the log to the opposite edge of the skillet. Repeat twice more with the remaining egg to form a large, loose roll.
Remove the skillet from the heat and set a lightly moistened wooden sushi mat over the omelet. Invert the omelet onto the mat. Bring the sides of the mat over the omelet and let rest for 5 minutes. Turn the omelet out onto a plate and thickly slice crosswise. Serve with daikon and soy sauce.