Heat 1 tablespoon olive oil in a nonstick skillet over medium-high heat. Add tomato, capers, and red pepper flakes; sauté until tomatoes just start to break down, about 3 minutes. Remove from the heat and stir in balsamic vinegar.

Brush one side of each bread slice with remaining olive oil. Spread the other sides with pesto. Arrange 2 slices of bread, pesto sides up, on a work surface. Layer with mozzarella cheese, followed by the tomato mixture, then sprinkle with salt and pepper. Top with the remaining bread slices, oiled sides up.