In a small saucepan, heat brown sugar and water over medium until sugar is dissolved. Remove from heat.
Stir balsamic vinegar, lime juice, and ½ cup basil into the sauce and let sit 20 for about minutes.
Cut each peach into 4 to 6 lengthwise planks, leaving on the skin and removing the pit.
Brush peaches with olive oil then grill over medium heat until charred and softened, about 5 minutes (you can also use a panini grill).
Cut mozzarella into thick slabs and season lightly with salt and pepper.
Arrange caprese stacks by layering a slice of peach, a few leaves of basil, a slab of mozzarella, and another peach. Secure with a toothpick if desired.
Strain basil leaves from the sauce and lightly drizzle over caprese stacks.