In a small saucepan, combine dashi stock, mirin, soy sauce, and sugar. Bring to a gentle simmer over medium heat, stirring until sugar dissolves. Remove from heat and set aside to cool.
Ensure all vegetables are thoroughly washed and patted completely dry. Excess moisture will prevent the batter from sticking.
In a large bowl, whisk the egg and ice-cold water until just combined. Gently sift the cake flour into the liquid mixture. Mix with chopsticks or a fork *very briefly* until just combined and still lumpy. Do not overmix, as lumps are desirable for a crispy tempura. Keep the batter cold.
Pour vegetable oil into a deep pot or wok, filling it about 5-7 cm deep. Heat the oil over medium-high heat to 170-180°C (340-350°F). You can test the temperature by dropping a small amount of batter into the oil; it should sink slightly then quickly float to the surface with gentle sizzling.
Dip each piece of vegetable into the cold batter, ensuring it's lightly coated. Gently lower a few pieces into the hot oil, ensuring not to overcrowd the pot. Fry for 2-4 minutes, turning occasionally, until golden brown and crispy. The frying time will vary depending on the vegetable. Sweet potato and eggplant might take slightly longer.
Using a slotted spoon or spider, remove the fried tempura from the oil and transfer to a wire rack lined with paper towels to drain excess oil. Serve immediately with the prepared Tentsuyu sauce, optionally garnished with grated daikon and ginger.
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