Place 3 cups of water and the cucumber chunks in a blender. Blend on high speed for about 30 seconds until the mixture has a smooth, even consistency.
1 large standard cucumber peeled and cut into large chunks, 4 cups water, divided
Using a fine-meshed sieve, strain the blended cucumber mixture into a pitcher. Use a spoon to stir the mixture in the sieve and ensure that the maximum amount of water drains through. Discard the cucumber pulp. See notes.
Place the strained mixture back into the blender and add in the remaining water, lime juice, peppermint leaves, and sugar. Blend on high until the mixture is smooth and no large chunks of peppermint leaves remain.
1/4 cup lime juice, 1/4 cup packed spearmint leaves, 3 tbsp granulated sugar, 4 cups water, divided
Taste for sweetness and add more if desired. I like to give this drink a quick whiz in the blender between each addition of sugar. Pour into a pitcher for serving.
Chill in the fridge for at least 2 hours or serve immediately over ice!
Although the mixture is strained, you may still see some settling after the drink rests. This is normal! Just give it a quick stir before you pour into individual glasses.