Scrub each lemon under running water and pat them dry. Slice off the ends from each lemon, so that they have flat tops and bottoms.
With a lemon standing on its flat end, quarter it with a knife from the top to half an inch from the bottom, leaving it attached on one end. Repeat for the remaining lemons.
Slightly spread open each lemon and add a tablespoon of kosher salt to each, making sure to cover the exposed flesh. Reshape the fruit.
Add another tablespoon of salt to the bottom of a glass jar. Add the lemons one or two at a time. Using a muddler or back end of a wooden spoon, pack them in and let the juice squeeze out to allow more room for the rest of the lemons (don't worry if any lemons break apart).
After you've added all the lemons to the jar, sprinkle with another thin layer of salt and squeeze them down some more. If the juice released doesn't cover the lemons, add more freshly squeezed lemon juice (do not use water or bottled lemon juice).
Close the jar and store in the refrigerator to ripen, shaking the jar every day for three to four weeks.
Aside from the lemons you're preserving, it's good to have a couple more handy in case you need more juice to top the lemons with. I always purchase 2-3 more depending on size. If you don't end up needing them to add to the jar, you can use them for plenty of other recipes or even cleaning your microwave.