With a large spoon, scrape out and discard honeydew seeds. Using a 3/4-inch melon baller, scoop out 10 balls; place in a large sealable plastic container. Pour Midori over melon balls; transfer to freezer, and let macerate about 1 hour.
Meanwhile, place remaining honeydew in a blender, and puree until smooth. Let stand until juice settles to the bottom of blender and foam rises to the top. Skim off foam, and measure out 1/2 cup juice.
Remove honeydew-ball mixture from freezer; stir in juice and seltzer. Divide evenly between two cocktail glasses; serve immediately.
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