Preparation

For the granita:

• Bring the sugar and 2 Tbs. water to a gentle boil in a small saucepan over medium-high heat, swirling the pan to dissolve the sugar; reduce the heat to medium and simmer for 2 minutes. Off the heat, cool the syrup to room temperature. Place the syrup, salt, 2 tsp. of the lime juice, and the cucumber in a blender and puree until smooth. Pour the cucumber mixture into a shallow container. Cover and freeze until softly frozen (not solid), 2-1/2  to 3 hours. Remove from the freezer and gently run a dinner fork through the frozen mass to break it down into loose, icy crystals. Cover the container and return it to the freezer. Freeze until firm, checking and raking the granita with the fork every hour or two to keep the crystals loose and fine, at least 7 hours.

For the shake:

• Place 1/2 cup of granita, the honeydew, and the remaining 1 tsp. of lime juice in a blender and blend to break down the honeydew completely, about 1 minute. Add the ice cream and pulse several times to begin breaking it up. With the blender motor off, use a flexible spatula to mash the mixture down onto the blender blades. Continue pulsing, stopping, and mashing until the mixture is well blended, thick, and moves easily in the blender jar, roughly 30 to 90 seconds. Pour into a chilled glass or glasses, top each with 1/2 cup granita, and serve at once.