Combine dashi, soy sauce, mirin, and sugar in a small saucepan and bring it to a boil. Then lower the heat and let it simmer until sugar is completely dissolved. Remove from heat and set aside.
To Prepare Ingredients
To make the shrimp straight, follow this instruction on how to straighten the shrimp. Pat dry and make sure moisture is completely removed from the shrimp.
Cut all the vegetables into thin 5-mm slices. For eggplant, discard the head, then cut it in half lengthwise. Followed by cutting the eggplant lengthwise into very thin slices (about ⅛ inch or 3 mm) leaving the top 1-inch part intact. Then, gently press down to fan the slices out.
To Prepare Oil
In a medium-size pot, heat 1 ½" to 2" (3-5 cm) of the oil to 350°F (180°C).
To Make Tempura Batter
While oil is heating up, start preparing the tempura batter. Add the egg into the icy water in a 2-cup measuring cup (or any bowl). Whisk the egg mixture vigorously and discard the foam on the surface.
Sift the flour into a large bowl. Slowly pour the egg mixture into the flour. Mix the batter but do not over mix; it's okay to leave some lumps in the batter. Keep the batter cold at all times (keep in the fridge). Make batter right before deep frying to avoid activation of wheat gluten.
To Deep Fry Tempura
Check the oil temperature with a thermometer to make sure the oil is 350°F (180°C). You can also use chopsticks; when you see small bubbles around chopsticks, it’s ready for deep frying. If you want to read more about deep frying tips, please read this post.
Start deep frying with cleaner ingredients (less astringent); for example, the cooking order would be shiso → mushroom → eggplant → shrimp → sweet potatoes, and kabocha. Coat the ingredients in the batter. For shrimp, sprinkle some flour or potato starch (or cornstarch) over to will help adhere to the tempura batter.
Deep fry the ingredients until golden brown. Do not crowd the pot because the oil temperature will drop quickly. Here's the rough cooking time for the ingredients. Shiso (350ºF/180ºC, 20-30 seconds)→ mushroom (350ºF/180ºC, 1 min) → eggplant (350ºF/180ºC, 1 min) → shrimp (350ºF/180ºC, 2 minutes)→ sweet potatoes, and kabocha (320ºF/160℃, 2-3 mins).
Transfer the Tempura to a wire rack or a plate lined with a paper towel to drain excess the oil.
Between batches, scoop up the crumbs in the oil (this is called Tenkasu), which will burn and turn the oil darker if left in the pot.
Grate the daikon and squeeze the liquid out. Prepare 3-4 Tbsp of warm Tentsuyu in individual small bowls and serve grated daikon on the side.
Put 1 Tbsp of grated daikon in the sauce and dip the tempura in the sauce to enjoy.