Pour oil into 2-qt. saucepan until depth of ½" is reached. Heat over medium-high heat. Add capers and cook, stirring occasionally, until popped open and crisp, 1-2 minutes. Using a slotted spoon, transfer to paper towels to drain. Working in batches, fry basil leaves until crisp, about 30 seconds. Transfer to paper towels to drain.
To serve, arrange tomato and mozzarella slices in an alternating pattern on a serving plate. Tuck the fried basil pieces between the tomato and mozzarella, season to taste with salt and pepper, and scatter with fried capers. Drizzle with vinegar and olive oil.