Place the lime quarters, lime juice, mint sprigs, sugar and [2 cups] water in a blender or food processor. PUREE until smooth. Strain the mixture through a fine mesh sieve into a pitcher.
Return the strained mixture to the blender or food processor, along with another [2 cups] cold water. PUREE again until smooth.
Strain again into the pitcher, then add the remaining 2 cups cold water; stir well. Chill the beverage before serving or you can serve it immediately over ice cubes (with mint leaves or sprigs as garnish, if desired).