In a large pitcher, combine the granulated sugar and cold water. Stir gently until the sugar mostly dissolves, then refrigerate to chill.

Rinse the limes to remove any dirt or wax, trim both ends, and cut into wedges.

Blend half of the lime pieces with 2 cups of chilled sugar water until rind breaks up.

Strain the lime mixture through a mesh strainer into a bowl, pressing to extract juice, and discard rinds.

Repeat with remaining lime wedges and another 2 cups of sugar water.

Blend strawberries with the final 2 cups of sugar water, then strain through a sieve to capture the fruity sweetness.

Combine all strained liquids back into the pitcher and stir in sweetened condensed milk until fully mixed.

Pour the refreshing lemonade over ice and garnish with lime slices or strawberry pieces.