Cut each shrimp partway along its back and open gently into a butterfly shape without separating the halves. Pat dry and season with salt and 1/4 teaspoon paprika.
Put flour, beaten eggs and panko in three separate shallow dishes. Coat shrimp in flour, egg and crumbs, then brush gently with the measured oil.
Preheat air fryer to 375°F. Cook eight shrimp at a time in one layer for 7 to 9 minutes, turning once, until golden and the thickest parts reach 145°F; repeat for three total batches. Keep cooked shrimp on a rack in a 200°F oven while finishing.
Stir mayonnaise, capers, lemon juice, mustard, hot sauce and remaining paprika into a remoulade. Split the rolls and lightly toast their cut faces under the broiler.
Spread remoulade inside each roll and add romaine and blotted tomato slices. Arrange six hot shrimp along each roll.
Close gently without crushing the crumbs and serve one sandwich per person immediately.