Pour balsamic vinegar and honey into a deep skillet over medium heat. Cook and simmer until reduced to slightly under a 2 Tablespoons, about 15 minutes, stirring occasionally.
Preheat oven to 500 degrees F. Brush flatbreads with olive oil. Layer with thinly sliced mozzarella cheese and thinly sliced peaches. Bake for 4-8 minutes or until cheese is melted and bubbly. Remove from oven. Sprinkle with chopped basil and drizzle with balsamic reduction.