Make the aioli. Combine the vegan mayo, lemon juice, minced garlic, salt, and pepper in a small bowl. Stir until smooth and pop it in the fridge to chill while you work on the olives.
Make the batter. Whisk half of the flour together with the water, paprika, garlic powder, and black pepper. You want a smooth, thick consistency that clings to the olives without sliding off.
Set up your breading station. Put the remaining dry flour on one plate, the panko on another, and keep the batter bowl in the middle.
Coat the olives. Take each olive through the three stages: toss in the dry flour first, then into the batter, then roll through the panko until fully covered. Set aside on a clean plate.
Get the oil hot. Fill a Dutch oven or deep pan with a few inches of oil and bring it to 325 to 350°F (170 to 180°C). Use a thermometer to keep it in range.
Fry in batches. Drop the olives in a few at a time, giving them space to float and crisp up evenly. They only need about 1 to 2 minutes until golden all over.
Drain and serve. Transfer to a paper towel-lined plate for a minute, then serve hot alongside the chilled garlic aioli.
Notes
Drain and pat the olives completely dry before breading for the best coating adhesion. The batter should be thick enough to coat the back of a spoon. If it’s too thin, add a bit more flour. Fry in small batches to keep the oil temperature steady. These are best served immediately, but you can bread them ahead of time and fry right before serving.