Honeydew melons appear in the produce section of grocery stores in mid to late spring, and have a milder, sweeter flavor than their cantaloupe cousins. Select a firm honeydew that feels heavy for its size and has unblemished skin. Store whole honeydew melons for up to three days at room temperature and up to five days in the refrigerator, or refrigerate cut-up honeydew for up to three days in a covered container.
Rinse the honeydew melon under cool running water, and scrub the entire surface, including both the blossom and stem ends, with a vegetable brush. Cut off and discard any bruised, overripe or rotting areas on the skin.
Place the cutting board on a firm surface; create a stable base, if necessary, by setting the board on a kitchen towel or other nonskid material.
Set the honeydew melon on the cutting board with its stem end pointing up. Use a large chef's knife to cut a thin piece off the right or left side of the melon, depending upon whether you are right- of left-handed. Turn the melon so that the flat base rests on the cutting board. Cut the stem and blossom ends off, and then cut the melon in half either crosswise or lengthwise.
Use a soup spoon or your hand to scoop out the seeds and membrane from the cavity of the melon, and discard them. Set one-half of the melon aside, and place the other half, cavity-side down, on the cutting board.
Cut the skin from the melon, starting at the top and working down to the cutting board, removing just what you can easily manage without having to bear down too hard on the knife. Remove the skin down to the green flesh, leaving no trace of the white outer flesh.
Cut the peeled melon into 1/2-inch-thick slices, either crosswise or lengthwise, and serve or refrigerate in a covered container. You can also cut the slices again into cubes about every 1/2 to 1 inch along their length and serve or refrigerate in a covered container. Repeat with the other half.