Use a melon baller to scoop out balls from the melons. You will need 18; set aside.
With a sharp knife, cut out very thin Parmesan slices. Triangles are fun if you can manage that (again, you will need 18). Set aside or store in a sealed container and chill until ready to use.
With a sharp knife, cut the prosciutto into 2½-inch squares. Cover the prosciutto loosely with plastic wrap to prevent it from drying out while assembling the appetizers.
On a clean work surface, place one melon ball in the center of one prosciutto slice, then fold all sides of the prosciutto over the melon ball as you would a package. Place the finished purse, seam-side-down, on a baking sheet and cover with plastic wrap. Repeat with remaining prosciutto/melon.
When ready to serve, arrange Parmesan slices on a platter. Place a prosciutto-wrapped melon ball on top of each Parmesan slice and serve.
Notes
You can prepare the prosciutto wrapped melon balls up to a day ahead. Keep tightly covered with plastic wrap and refrigerate.
For a pretty serving idea, scatter whole chives on a platter and place appetizers on top of the chives.
If preparing this recipe as GLUTEN FREE, just be sure to use a brand of prosciutto that is known to be GF.