Chop green onions thinly and grate the daikon radish. Set aside in the fridge. Thinly cut the vegetables, and cut shrimp into small pieces. Put the vegetables and shrimp in a medium bowl.
For the Tempura batter, mix flour, Katakuriko or cornstarch, a pinch of salt, and water in a different bowl. Do not overmix. Add vegetable to the batter.
Fill a pan with oil to 1" (2.5 cm) deep and heat over medium-high. Drop big spoonfuls of the battered vegetables into the oil and fry for 6–7 minutes, turning once or twice. Remove from oil.
Cook Udon according to the package instructions, and cool under running water. Use ice if needed to cool further.
Mix Mentsuyu and water. In large shallow bowls, divide the noodles, Kakiage tempura, and green onion and radish toppings. Pour the sauce over the dish.