In a small bowl, stir the soy sauce, vinegar, sesame oil, ginger, and brown sugar until the sugar is dissolved.
Add enough vegetable oil to a 14-inch flat-bottom wok (preferably seasoned carbon steel) to measure 1-1/4 inches deep; heat over high heat until the oil registers 365°F on a deep-fry thermometer.
While the oil heats, whisk the flour, beer, and 1 tsp. salt in a medium bowl until smooth. Working in batches of about 10 pieces, add vegetables to the batter and stir until just coated.
Using wooden chopsticks or tongs, carefully add the first batch of battered vegetables, one at a time, to the oil. Fry, turning the pieces halfway through cooking, until light golden, 1 to 2 minutes. Using a metal skimmer, strainer, or slotted spoon, transfer the vegetables to a plate lined with paper towels. Repeat in batches of 10 vegetables until all are fried, adjusting the heat as needed to maintain the oil temperature. Serve immediately with the soy-ginger dipping sauce.