Mix equal parts chopped lemons with sugar in an appropriately-sized bowl or mason jar. Mix in the ginger. Cover (with lid or saran wrap) and refrigerate for 24-48 hours. Once it's macerated, the juice of the fruit and the sugar will form a syrup.
Prepare two vessels – one bowl (ideally a large 6-8 cup measuring cup) with a fine metal sieve, and the other (ideally a 2-cup measuring cup) draped with a 5-inch by 5-inch double-layered piece of cheesecloth.
This is the messy part: Working in batches, strain the syrup through the fine metal sieve and into the bowl, adding any undissolved sugar back into the mix. As you work, transfer the sugar-coated lemon pieces to the center of the prepared cheesecloth. Once all the syrup is transferred, gather the lemon pieces up in the cheesecloth, making it into a pouch. Squeeze the pouch thoroughly, getting out as much juice as possible from the fruit into the small bowl. Discard the pouch and fruit, and combine the small bowl of juice with the larger bowl of syrup. I had about 2 cups of syrup, but yours may differ.
Measure out an equal amount of vinegar (half apple cider and half white vinegar) to the amount of syrup you have, and combine in an airtight bottle or jar. Stir in 4 pinches of salt per total cup of the final mixture. The shrub will keep in the fridge for 2-3 weeks.