Brush olive oil on one side of each slice of bread. Place bread slices oil side down on a work surface.
Spread aioli on four of the slices of bread and arrange the tomatoes, mozzarella cheese, and basil on top of aioli. Top with remaining slices of bread, oil side up.
Preheat panini grill, or large skillet over medium heat. Cook sandwiches 5-8 minutes, flipping the sandwich over if cooking in a skillet, or until golden brown on both sides and the cheese has melted. Serve immediately