Mash lemons, mint leaves, and sugar (and salt, if using) in a large, deep bowl with potato masher or back of wooden spoon until lemon slices give up their juice, sugar is dissolved, and juice is thickened to syrup consistency, about 4 minutes.
Pour half the lemon slices and syrup through a large sieve over bowl; press on solids with masher or back of wooden spoon to release as much liquid as possible.
Discard solids; move liquid to serving pitcher.
Repeat process with remaining lemon slices.
Stir in water until blended.
Chill well and stir to blend before serving, over ice. Enjoy!