Cut ends off lemons. Standing one lemon on end, cut in quarters lengthwise, leaving bottom section in tact. Repeat with remaining lemons.
Scoop kosher salt in the center of a lemon and transfer to quart jar. Repeat with remaining lemons, pressing each in with the back of a spoon. Cover and let stand in a cool, dark place at room temperature overnight. The next day, press lemons down again with the back of a spoon and top with additional kosher salt to cover.
Transfer to refrigerator for one month, pressing down lemons occasionally to keep them below the brine. After the month, keep refrigerated and use as needed for up to one year*.