Step 1Prepare the melon: Cut it in half lengthwise and scoop out the seeds. Cut each half in half again lengthwise. Cut the tips from the ends of each quarter, leaving a chunk of melon with an interior that is roughly flat.
Step 2Stand 1 quarter of a melon on its end and carefully cut away the peel. Don't try to save too much -- the melon closest to the rind doesn't have much flavor. Cut 1 slice from the center to leave it perfectly flat. Carefully cut the melon that remains into 3 slices, each about one-fourth-inch thick. Use a sharp knife and a slow, even stroke to do this. You should have a rectangular sheet of melon of even thickness. Repeat with the remaining quarters.
Step 3Lay a sheet of plastic wrap on the work surface and position 1 melon layer in the center. Cut 1 slice of prosciutto in half and arrange it over the top to cover as much melon as possible. Trim and fit pieces as necessary. Cover this with another melon layer and repeat with another slice of prosciutto. Cover with a third melon layer. Wrap tightly in plastic wrap and press down firmly on top. Repeat with a second quarter and refrigerate at least 1 hour.
Step 4With the remaining 2 stacks, lay a sheet of plastic wrap on the work surface and position 1 melon layer in the center. Arrange 3 small basil leaves on top and cover with another melon layer. Arrange 3 more basil leaves on top, preferably pointing in a 90-degree angle to those on the first layer. Cover with a third melon layer. Wrap tightly in plastic wrap and press down firmly on top. Repeat with the remaining quarter and refrigerate at least 1 hour.
Step 5When ready to serve, unwrap a melon stack. Trim the rough edges to make each side perfectly even. Cut the stack in half lengthwise, then in half crosswise; cut each crosswise half in half again. This will result in 8 bite-sized pieces. Skewer each with a toothpick and arrange them on a serving platter. Repeat with the remaining melon stacks.