This is a new way to serve the classic combination of prosciutto with melon, one of Italians’ famously favorite ways to start a meal. You could pierce the prosciutto purses with toothpicks to make them easier to eat. And to make the preparation easier, be sure to place the block of Parmesan in the freezer for a few minutes, so that when you shave it, it won’t crumble.
Gently Scrape Out the Seeds and membranes from the cantaloupes and honeydews. Using a melon baller, scoop out balls from the flesh and place the melon balls in a large bowl. Discard the skin.
Trim the prosciutto slices into 2½-inch squares. Cover the prosciutto loosely with plastic wrap to prevent it from drying out while assembling the purses. Place one melon ball in the center of one prosciutto slice, then fold all the sides of the prosciutto over the melon ball as for a package. Place the finished purse, seam side down, on a baking sheet and cover with plastic wrap. Repeat with the remaining prosciutto squares and melon balls. (The prosciutto balls can be prepared up to this point 1 day ahead. Keep tightly covered with plastic wrap and refrigerate.)
Using a vegetable peeler and pressing firmly, shave the Parmesan into long slices (the cheese will crumble if too little pressure is used). Decoratively arrange the Parmesan slices on a large platter. Place a prosciutto purse atop each Parmesan slice and serve.