Place first 5 ingredients in a blender or food processor.
Blend or pulse until the shallot and basil are finely chopped.
Cut melon in half and scoop out the seeds. Using a small melon baller (about the same size as the cheese balls), create as many melon balls as you can.
Create your masterpiece! To arrange your skewer, start with a melon ball. Next ruffle a piece of prosciutto and slide that on. Next add a mozzarella ball then another piece of ruffled prosciutto and finally another melon ball. Continue until all skewers have been created. Arrange on your platter.
You can make this a couple of hours ahead but you should cover and place in the refrigerator. (Do not put the basil oil in the refrigerator).
When ready to serve, drizzle skewers with the basil oil and garnish with a few basil leaves.