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Cut the honeydew in half and then remove and discard the seeds using a metal spoon. Wrap up half of the melon and put it in the refrigerator to eat at your convenience.
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Cut the honeydew into 2 inch thick slices and then peel the outer skin off of each slice. It is better to cut them into long slices before peeling because they are easier to hold and peel that way than when they are still in a giant awkward half-sphere. Once all of the slices are peeled, cut the honeydew melon into roughly 2 x 3 inch chunks. Place half of the pieces in the freezer, and the other half in the refrigerator. Let them chill for 1 hour.
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Remove the honeydew pieces from the freezer and refrigerator and toss them together in a bowl so they are evenly distributed. If you have a large food processor, you don't need to do the following and divide everything in half and do it twice. I have a regular sized blender and it doesn't fit everything in it at once, so I had to divide the ingredients in half and do it in two batches, as follows.
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Place half of the the melon pieces into a blender with half of the other ingredients. Put the lid on the blender and blend on high until the mixture is completely blended, about 45 seconds. Pour the mixture into glasses, then repeat this process with the other half of the ingredients. Serve immediately with fun straws and a small slices of fresh honeydew, if desired. Makes about 4 servings.