In the container of a large 2-quart pitcher or large 64-ounce Mason jar, squeeze the juice of the lemons removing any seeds. Add 4 rosemary sprigs along with 4 slices of the lemon. Add the sugar and fill the pitcher with water and stir. Let steep for 2 hours or overnight.
In 4 tall glasses, rub each rim with 1 lemon slice. Invert the glass onto a plate covered with kosher salt and move the rim around the salt until coated.
In each glass filled with ice, add 1 jigger of rum and fill with lemonade. Finish by adding 1 slice of lemon and 1 sprig of rosemary for garnish. Repeat for each cocktail.
Notes
Look for Bayou Rum products wherever liquor is sold, and if they don't have it, ask for it. I like Meyer lemons for this, but regular lemons will work fine. Our lemon tree produces more lemons than we can use, so we have a "squeezing and freezing" party to enjoy the juice year round. You can make this drink by the pitcher full or individually in a cocktail shaker; either way, it works. I like the salt on the rim for an unexpected contrast of taste and texture; feel free to leave it out.
Recipe by Acadiana Table at https://www.acadianatable.com/2017/06/26/lemon-rosemary-rum-cooler/