Cut a small slice from 1 end of each lemon; stand lemons upright. Cut each lemon lengthwise into quarters, cutting to, but not through, the opposite end. Sprinkle 1 teaspoon kosher salt into each lemon. Place the lemons in a medium bowl; sprinkle with remaining kosher salt, juice, 1/4 cup water, and olive oil. Let stand, partially covered, at room temperature 2 days, stirring occasionally and pressing the lemons to extract juice. Cover and refrigerate 3 weeks, stirring occasionally and pressing to extract juice from lemons.
Rinse lemons; remove seeds and white pith. Chop peel. Place chopped peel in a small bowl; cover and refrigerate up to 6 months.