First, wash and dry cucumber, grapefruit, and basil leaves.
Second, make simple syrup:
Combine ½ cup of sugar and ½ cup of water straight into the bottom of the glass serving pitcher. (Making the “simple syrup” at the beginning of this recipe allows the sugar time to dissolve. A combination of equal parts sugar and water makes a syrup- “simple syrup”.) Next, slice grapefruit and squeeze out ALL the juice into a measuring cup, small glass, or pitcher.
Take your time here and extract all the juice from the grapefruit, set the juice aside. (If you have some grapefruit flesh left in the rind, you can cut a notch in the quartered grapefruit and use it to wet the serving glass rim before rimming the serving glass—instructions below.) Then cut ⅓ of the cucumber into thin slices and set aside the cucumber slices.
Peel the remaining ⅔rds of the cucumber. Cut the peeled cucumber into chunks.
Put the peeled cucumber chunks into the blender with ¼ cup of water to aid the blender in pureeing the cucumber.
Now pour cucumber puree straight into the pitcher (Do Not Strain Puree).
Then, strain grapefruit juice with a mesh sieve strainer into the serving pitcher.
Next, add the 10 oz Schweppes (or sparkling water of choice) and the remaining 1 cup of water (if you used ½ cup in the simple syrup and ¼ cup in the puree).
Stir all the ingredients together.
Chiffonade the basil: stack the basil leaves, roll the basil leaves up from the long side, and slice them into very thin slices. Set aside basil chiffonade.
Add crushed ice to top off the pitcher.
Garnish with the reserved cucumber slices and basil chiffonade.