Start by preparing your vegetables. Cut them into uniform, bite-sized pieces and pat them completely dry with paper towels. Any moisture will affect the batter's crispiness.
Fill a heavy-bottomed pot with about 2 inches of oil. Heat to 375°F (190°C). Use a thermometer for accuracy – temperature control is crucial for perfect tempura.
While the oil heats, prepare your batter. Keep all ingredients very cold. Mix the egg with ice-cold water in one bowl. In another bowl, combine flour, cornstarch, and salt.
When ready to fry, quickly combine the wet and dry ingredients. Mix just until barely combined – lumps are good! Overmixing will develop gluten and make the coating heavy.
Dip vegetables one at a time into the batter, letting excess drip off. Carefully place in hot oil.
Fry in small batches for 2-3 minutes until light golden brown, turning once. Don't overcrowd the pot.
Remove with a spider or slotted spoon and drain on a wire rack over paper towels.
For the dipping sauce, combine all ingredients in a small pot, bring to a simmer, then remove from heat.