Whisk the egg in a small bowl. Add flour and the cold water, and whisk to combine. Cover with plastic wrap and chill in the refrigerator.
To make the tempura, slice the eggplant in half lengthwise, then cut the bottom into 5 slices (be careful not to cut all the way through to the top).
Add flour to a shallow dish. Coat the eggplant, pumpkin, green bell pepper, and shrimp in the flour, shaking off any excess.
Heat the oil in a large pot until it reaches 350°F (180˚C).
Dip the floured vegetables and shrimp into the batter.
Fry the battered shrimp and vegetables until they float to the top and are golden brown in color. Use a slotted spoon to transfer them to a paper towel-lined plate to drain.
To make the kakiage, put the onion, carrot, Japanese parsley and dried shrimp into a medium bowl. Add 2 tablespoons of flour and toss to coat. Pour the leftover tempura batter on the top and mix until evenly coated.
Bring the oil in the pot back to 350°F (180˚C). Drop the kakiage by the spoonful into the oil and fry until golden brown in color. Use a slotted spoon to transfer them to a paper towel-lined plate to drain.
In a medium pan, combine the soy sauce, mirin and sugar, and bring to a boil over high heat for 2 minutes.
Top the rice with kakiage and tempura and drizzle with the sauce.