Chef Ouita Michel talked about these lemons—a "waste-not" home kitchen condiment—on the 10-27-15 episode of Hot Water Cornbread: Kentucky Food Radio.

Ouita says:

We use lemon halves that we have squeezed most of the juice out of for our lemon juice. I put 1 1/2 tsp. of kosher salt in each lemon half and pack them tightly into a gallon plastic jug-- a sterilzied quart jar would work fine at home. Stack the halves like little cups. After the last lemon is stacked up, we add a little more salt, and about 1 cup of lemon juice and olive oil to the jug. We flip the jar up side down ever day. Store for three weeks. Rosemary, bay leaf, cardamom, cumin corriander, tumeric, cinnamon can all be added for variations. 

When we use these kinds of preserved lemon, we remove the pith and use the skin, which should be nice and soft. You can puree it and use it as a seasoning or it can be julienned and added to dishes.