In a small saucepan over medium-high heat, melt butter then whisk in honey, garlic, red pepper flakes, and hot sauce and bring to a low simmer. Remove from heat and keep warm (if hot honey solidifies while it sits, gently reheat over low heat)
Preheat a grill to 400F. Brush peach halves well with olive oil.
Place peaches cut-side down on preheated grill and let cook 3-5 minutes until grill marks appear and peaches are slightly caramelized. Transfer peaches to a cutting board and cut each peach half into thin slices.
Arrange mozzarella on flatbreads and season with salt and pepper.
Lower grill heat to 350F. Slide flatbreads onto grill and immediately close grill. Allow flatbreads to cook 3-5 minutes until mozzarella is melted and bottom of flatbreads are slightly charred. Use a spatula to slide flatbread off of grill and onto a cutting board.
Top flatbread with grilled peach wedges, drizzles of hot honey, and fresh basil. Cut into wedges and serve immediately while warm. Enjoy!