Combine the granulated sugar and cold water in a large pitcher. Stir gently until most of the sugar is dissolved, then chill in the fridge for about 30 minutes.

Rinse the thin-skinned limes and trim both ends. Cut them into wedges or eighths.

In a blender, add half of the lime pieces and 2 cups of the chilled sugar water. Pulse until the rind breaks up but do not over-blend.

Strain the blended lime mixture through a mesh strainer into a bowl, pressing to extract juice. Discard leftover rinds.

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

Blend the strawberries with the final 2 cups of sugar water and strain to capture the fruity sweetness.

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

Pour the lemonade over ice and garnish with lime slices or fresh berries as desired.