Make the batters: In a medium bowl, add the flour. In a separate medium bowl, whisk the egg and 1⅓ cups room-temperature water to combine. Skim away any foam from the surface and discard.
Measure 1⅓ cups of the egg mixture, and set aside the rest. Working in 3 batches, gradually pour the 1⅓ cup of the egg mixture into the flour, whisking just to incorporate after each of the first 2 additions. Before the last addition is completely mixed in, tap the whisk against the side of the bowl several times to dislodge any flour. Switch to thick cooking chopsticks (konabashi), and stir about 18 times in a figure-eight formation. (Do not overmix; the perfect batter should still be dotted with a few small flour lumps.)
Add half the batter to a second medium bowl. In one (this will be the kakiage batter), stir in the reserved egg mixture.
Fry the tempura, serving and eating as you go: Place a wire rack inside a rimmed baking sheet, and set it by the stove. In a deep, 10-inch copper, ceramic, or stainless-steel pan, add enough canola oil to reach 1¼ inches up the sides. Heat until the temperature registers 355°F on a deep-fry thermometer.
Meanwhile, in a medium bowl, add the flour. Match up 8 pairs of similar-size shishitos, and use a toothpick to skewer each pair together just below the stem. Using the thick chopsticks, dip one side of the peppers in the flour, shake off any excess, then dip the floured side into the tempura batter (not the kakiage batter). Shake off any excess. Transfer to the oil batter-side down and, working quickly, repeat with 3 more pairs. (The oil will drop to 340°F; maintain this temperature during frying.) Let cook for 40 seconds, then turn by nudging gently with a pair of thin serving or cooking chopsticks (moribashi or saibashi). Let cook 20 seconds more on the remaining side, or until the batter is crispy. Remove using a spider skimmer or the thin chopsticks. Gently shake off excess oil, and drain briefly on the rack. Serve all the tempura hot with Tempura Dipping Broth, curried salt, or salted lemon juice.
Fry the eggplant: Trim the ends and caps. Halve the eggplants lengthwise and place cut-side down on a cutting board. Holding your knife lengthwise and placing the tip ⅓ of the way from the bottom of the eggplant, slice all the way through the rest of the eggplant at ⅛-inch intervals. Press down gently so the pieces fan out slightly but not so much that flour or batter gets in. Working with one at a time, dredge in the flour, shake off excess, then lightly coat in tempura batter, shaking off any excess. Using the thick chopsticks, transfer up to 2 pieces skin-side up to the oil at 355°F. The oil will drop to 340°F; maintain this. Cook for 20 seconds, then carefully turn with the thin chopsticks. Turn again, cook for 20 seconds, and turn back. Cook, turning once per minute, until tender and the batter is crispy, 3–4 minutes. Shake off excess oil and place on the rack. Serve hot.
Fry the green bean kakiage: Toss the beans in the flour to lightly coat. Using the thin cooking chopsticks, grasp 10 parallel beans into a bundle, shake off excess flour, and lightly coat with the kakiage batter, shaking off excess. Use a slotted ladle or large fork to transfer to oil at 355°F. The oil will drop to 340°F; maintain this. Let cook without turning until crispy, 1½ minutes. Remove to the rack. Repeat with the remaining green beans. Serve hot.
Fry the squash: Trim a 1-inch triangular piece from the top and bottom tip of each piece, and remove any hard corners. Line the wire rack with 2 layers of thick paper towels. Flour the squash, shake off any excess, and lightly coat in the tempura batter, shaking off any excess. Using the thick chopsticks, transfer up to 2 pieces into the oil at 355°F. The oil will drop to 330°F; maintain this. Cook, turning once per minute, for 10 minutes. Then cook for 3 minutes more on each side. Transfer to the rack, shaking off any excess oil, and wrap tightly with the paper towels. Let rest for 10 minutes. Unwrap, use the back of the knife to knock the batter from the skin side of the squash, then halve the wedges vertically. Serve hot.
Fry the corn kakiage: Coat the kernels lightly in the flour. Using a slotted ladle or spider skimmer, transfer ¼ cup of the kernels at a time to the kakiage batter, shaking off excess flour. Remove from the batter and shake off any excess. Transfer to the oil at 355°F, and repeat to fry 4 scoops at once. The oil will drop to 340°F; maintain this. Fry until the batter sets and the kakiage floats, 2 min-utes. Divide the remaining kernels across the top of each fritter, and cook 1 minute more. Use the thin chopsticks and the side of the pan to gently turn the pieces. Cook 1 minute more, then remove to the rack. Serve hot.