In a food processor combine half of the melon chunks and the 1/2 cup basil. Cover and process until smooth. Set a large sieve over a large bowl. Pour pureed melon into sieve. Repeat with the remaining melon chunks; pour into sieve. Cover and place in the refrigerator about 1 hour to allow melon mixture to drain.
Using the back of a large spoon, press melon pulp through sieve, draining additional liquid (you should have about 3 1/2 cups liquid). Discard solids.
For each drink, fill a cocktail shaker three-fourths full with ice. Add 1/3 cup melon liquid. Cover and shake until the outside of the shaker becomes frosty. Strain into a chilled glass. Slowly pour in 1/3 cup Prosecco; stir gently. If desired, add a melon slice and/or additional basil.