Preheat the oven to 375 degrees Fahrenheit.
Flatten the sheet of puff pastry and cut each sheet into 6 equal rectangles.
To build one pastry: rip one piece of prosciutto apart into 3-4 equal-sized portions. Layer those pieces of prosciutto in the center of one rectangle of puff pastry, leaving about 1/2 inch border. Layer 4 peach slices on top.
Line 2 sheet pans with parchment paper. Place the pastries onto the pan and bake for 25 minutes, rotating the pan after 10 minutes to ensure even browning.
Remove the pan from the oven and let the pastries cool for 5-10 minutes.
Slice the burrata open and separate into approximately 1 tsp size portions. Top each tart with as much burrata as you please. Add basil and fresh cracked pepper on top. Finish them off with a generous drizzle of honey and enjoy!