Heat the oil. Pour 2 inches of oil into a heavy-bottomed Dutch oven and clip on a thermometer. Heat over medium until the oil reaches 350°F and hold it there.
Dry and dredge the shrimp. Pat the shrimp completely dry with paper towels. Toss with ¼ cup flour in a shallow bowl, then transfer to a strainer and shake off excess.
Mix the batter. In a medium bowl, whisk together the flour, cornstarch, baking powder, baking soda, sugar, salt, paprika, onion powder, garlic powder, and white pepper. Pour in the ice-cold club soda and whisk just until smooth — a few small lumps are fine. Use within 5 minutes.
Fry in small batches. Dip each shrimp into the batter, let drip for one second, and lower into the hot oil. Fry 5 to 6 shrimp at a time for 2 to 2½ minutes, flipping once, until deep golden and crackling.
Drain and salt. Transfer to a wire rack set over a sheet pan. Sprinkle with flaky sea salt while still glistening. Serve within 5 to 8 minutes for peak crispiness.