In a large pitcher, combine the sugar and cold water. Stir gently until most of the sugar dissolves, then chill this mixture in the fridge while you prepare the other ingredients.

Rinse the limes thoroughly to remove any dirt or wax. Trim both ends and cut them into wedges or eighths to facilitate blending.

Add half of the lime pieces to a blender with 2 cups of the chilled sugar water. Pulse several times, just enough to break up the rind without making it bitter.

Pour the blended mixture through a mesh strainer set over a bowl. Press gently with the back of a spoon to extract as much juice as possible and discard the leftover rinds.

Repeat the blending and straining process with the remaining lime wedges and another 2 cups of sugar water to keep the juice smooth and free of pulp.

Place the strawberries into the blender with the final 2 cups of sugar water. Pulse briefly, then strain through a mesh sieve just like with the limes to add a sweet, fruity layer to the drink.

Combine all the strained liquids back into the pitcher. Stir in the sweetened condensed milk until fully incorporated.

Pour the lemonade over ice and garnish with lime slices or berry pieces for a refreshing, flavorful finish.