Place the eggplant wedges in a large bowl and toss with kosher salt and lemon juice. Allow to sit at room temperature for 1-2 hours, loosely covered with a dishcloth. Drain the eggplant in a colander and rinse in cold water. Gently press the eggplant to squeeze as much liquid out as possible, without smashing it.

Bring the vinegar to a boil in a wide saucepan. Add the drained eggplant, and return to a boil over medium-high heat. Cook the eggplant in the boiling vinegar for 2 minutes, then transfer to a bowl with a slotted spoon. (Keep the vinegar at a low simmer.) Add the torn basil, minced garlic, and black pepper to the eggplant, and toss well to combine. Scoop the eggplant into hot sterilized pint jars (see note), and top with the hot vinegar, leaving 1/2 an inch of headspace. Tap the jars to remove air bubbles and wipe the rims. Apply heated lids and rings. Process the pickles in a boiling water bath for 10 minutes, starting the 10 minutes after the water returns to a boil after adding the jars. Transfer to a folded towel and allow to cool. After 1 hour, remove the rings and check the seals. After 1 hour, check the seals by removing the bands from the jars, then holding the jars by their lids. The lids should hold firm. 

These pickles need to cure for about a week before enjoying them. Store sealed jars in a cool, dark cupboard.