Makes just over 1 quart (4 cups). Serves 4 to 6 and up to 8 if you like a lot of fizzy water in yours.
1 pound cucumber(s), peeled or unpeeled, cut into large chunks, plus a few extra thin cucumber slices for garnish
1 cup lemon juice (from about 7 to 8 lemons, although juiciness will vary)
1/3 cup granulated sugar
2 cups cold water
Run cucumber through a blender or food processor until pureed, then run it for a full extra minute to ensure that it’s as processed as possible. Set a fine-mesh strainer or a regular strainer lined with a couple layers of cheesecloth or a coffee filter over a pitcher and pour cucumber puree through it, stirring to help it move along faster. Discard solids. In the pitcher, you should have about 1 cup cucumber juice. Add lemon juice and sugar to it, then water. Give it a good stir or shake, and let it sit in the fridge (to get it started chilling) for 15 minutes, after which a couple more stirs or shakes should leave the sugar fully dissolved. Taste lemonade, adding more sugar or water if desired. (See Notes up top about concentration and sweetness.) Serve chilled over ice, with or without a splash of seltzer on top. You can fancy up the glass with a garnish of cucumber slice, if desired.