Cut all the vegetables into thin, even pieces — thickness is what decides whether the inside cooks through before the batter over-browns.
Whisk the flour, cornstarch, baking powder and salt together dry, then pour in the ice-cold water and stir just until combined, leaving a few visible lumps.
Heat the oil to 170 to 180°C / 340 to 355°F in a deep, heavy pot — test with a drop of batter, which should sizzle and rise within a couple of seconds.
Dip each piece of vegetable or shrimp in the batter, letting the excess drip off, and lower it into the oil away from you.
Fry in small batches of 4 to 5 pieces, 3 to 4 minutes, until golden and the bubbling around them has visibly slowed.
Lift out with a slotted spoon or spider onto a wire rack, not paper towels, so steam escapes from underneath instead of softening the bottom.
Let the oil come back up to temperature between batches before adding the next one.
Stir together the soy sauce, rice vinegar, scallion and gochugaru for the dipping sauce and serve it alongside while the twigim is still hot.