Dry the olives and mix the aioli. Drain the olives and pat them thoroughly dry, including around the filling. Stir the vegan mayonnaise, lemon juice and minced garlic together. Taste before adding salt and pepper, then refrigerate while you prepare the olives.
Make the batter. Whisk 1/4 cup of the flour with the cold water, paprika, garlic powder and black pepper until smooth. It should leave a thin, even coating on an olive. If it is too stiff, add cold water a teaspoon at a time.
Set up the breading station. Put the remaining 1/4 cup flour in a shallow bowl and the panko in another. Keep the batter bowl between them.
Coat the olives. Toss each dry olive in flour and shake off the excess. Dip into the batter, let the excess drip off, then roll in panko and press gently to cover. Set on a clean plate.
Heat the oil. Add about 2 inches of canola or vegetable oil to a deep, heavy pot, keeping the pot no more than one third full. Heat to 350°F (175°C), checking with a frying thermometer.
Fry in small batches. Carefully lower the breaded olives into the oil with a slotted spoon. Fry for about 1 to 2 minutes, turning gently, until golden all over. Let the oil return to 350°F (175°C) between batches.
Drain and serve. Lift out with a slotted spoon and drain briefly on paper towels. Let the centers cool slightly, then serve warm with the chilled garlic aioli.
Notes
Olives: Measure 5.5 oz (155 g) after draining. Use pitted olives and dry the surface carefully before breading.Aioli: The garlic is bold. Start with less if you prefer a milder dip and add more to taste.Make ahead: Refrigerate the breaded olives in a single layer for up to 4 hours. Fry just before serving. Refrigerate leftovers and aioli separately for up to 2 days; reheat olives in an oven or air fryer until hot and crisp.Nutrition estimate: Per quarter of the recipe, including all of the aioli and coating ingredients. The calculation allows for 2 tablespoons of frying oil absorbed across the whole batch. Actual oil absorption and unused breading vary, so these are estimates rather than measured values. Optional added salt is not included; sodium varies with the olives and mayonnaise.