In a small saucepan over medium heat, bring balsamic vinegar and blackberries to a boil. Continue to cook until the vinegar thickens, approximately 3-5 minutes. Strain out blackberries and stir in brown sugar. Set aside.
Preheat oven to 375 degrees. Add pizza crusts to a baking sheet and bake for 7 to 8 minutes. Remove from oven.
While the crusts are baking, heat up a grill or grill pan. Once hot, add sliced peaches to the grill or grill pan. Cook for 1 to 2 minutes per side. Remove from grill.
Layer grilled peach slices, blackberries, prosciutto, arugula, and burrata.
Drizzle over blackberry balsamic vinegar. Serve.