Wash the limes really well, making sure to scrub the peel well.
Cut 4 of the limes into 8 pieces each. Leave the 5th lime whole for now.
To the jar of a blender, add 16 pieces of cut limes, 1/2 cup of sugar, and 2.5 cups water (this is half of each ingredient).
Pulse the blender 7-10 times, just enough to break up the limes really well without completely pureeing them. You want the limes in little pieces, but the more you blend the limes the more bitter the juice will be.
Place a fine mesh strainer over a large bowl (or a pitcher with an opening wide enough to handle your strainer) and strain the mixture, letting it strain while you blend continue working.
Add the remaining lime pieces, sugar and water to the blender and repeat.
Pour the mixture through the fine mesh strainer.
Use a wooden spoon to gently press the mixture to get as much liquid out as you can. Toss the solids.
Carefully transfer the mixture to a pitcher.
Stir in the sweetened condensed milk.
Slice the remaining whole lime into thin circle slices. Use your knife to cut a small slit into one side of each circle and place on the edge of the glass as a garnihs.