4 cups water, divided use
1/2 cup sugar, plus more, to taste
1 English cucumber, peeled and roughly chopped
1/4 cup lime juice, plus more, to taste
Spearmint sprigs, for garnish (optional)
Ice cubes

Blend 3 cups of water with the sugar in a blender until the sugar has completely dissolved, about 30 seconds. Add the cucumber and lime juice and blend again until it is very smooth. Strain the mixture through a sieve and into a large container, and then stir in 1 cup of water. Gradually season the agua fresca to taste with more sugar and lime juice. Chill the pitcher in the refrigerator, then stir, thoroughly, add the mint sprigs, if using them, and pour the agua fresca into ice-filled glasses.