2 medium Persian cucumbers or 1 English hothouse cucumber, peeled, coarsely chopped
1/3 cup (packed) mint leaves, plus more for serving
Pinch of kosher salt
3 tablespoons fresh lime juice
Club soda (for serving)
Directions
Purée honeydew, cucumbers, mint, and salt in a blender until smooth. Strain through a fine-mesh sieve into a large bowl (do not press on solids, which would make mixture cloudy). Stir in lime juice and chill 1 hour.
Divide honeydew purée among ice-filled glasses. Top off with club soda and garnish with mint sprigs.
Honeydew purée can be made 1 day ahead. Cover and chill.