In a large pitcher, combine cold water and granulated sugar, stirring vigorously until the sugar dissolves completely. Allow the mixture to chill in the refrigerator for about 30 minutes.

While the mixture is chilling, wash the limes and cut them into eighths.

Blend half of the lime wedges with half of the chilled water and sugar mixture for 5 pulses.

Strain the blended mixture into the chilled pitcher using a fine mesh sieve.

Repeat blending with the remaining lime wedges and the remaining sweetened water mixture.

Stir the pitcher gently to combine before adding ice and sweetened condensed milk. Mix well.

Serve immediately in chilled glasses, garnished with a lime slice.