Gather ingredients:

Slice prosciutto: Carefully peel 4 prosciutto slices apart and place on cutting board in an even layer (reserve remaining prosciutto in refrigerator until ready to use). Using a very sharp knife, halve each slice lengthwise to create 8 long strips.

Roll prosciutto strips: Roll 2 (12- x 18-inch) sheets of parchment paper to form 2 (1 ¼-inch diameter, 12-inch long) cylinders; seal ends with paper clips to prevent them from unraveling. Carefully wrap prosciutto strips around parchment cylinders, arranging strips 1-inch apart with seams aligned (4 strips per cylinder). Remove paper clips.

Microwave prosciutto: Place, seam-side down, on a microwave-safe plate lined with 2 layers of paper towels. Microwave on HIGH for 40 seconds. Rotate cylinders, then continue microwaving on HIGH in 10-second intervals, until prosciutto begins to crisp, about 20 to 30 seconds more. Transfer to a clean paper towel-lined plate. Let cool slightly, 1 to 2 minutes. Slide crispy prosciutto rings off parchment cylinders and let cool completely, about 5 minutes. Discard used parchment cylinders. Repeat with remaining 4 slices prosciutto.

Make hot honey sauce: Cut peaches into 16 even wedges (about ½-inch thick). Whisk together honey, vinegar, hot sauce, and pepper in a small bowl until smooth.

Assemble: Carefully insert 1 peach wedge into each prosciutto ring. Brush 1 side of wrapped peaches with honey mixture and transfer to a serving platter, brushed side up. Drizzle with any remaining honey mixture and sprinkle with mint. Serve with goat cheese.