Trim the cabbage and slice it into thin ribbons about 1/8–1/4 inch thick. Place half of the shredded cabbage into a medium heavy-bottomed pot.
Grate one half of the apple directly over the cabbage so the juices fall in. Stud the remaining apple half with the whole cloves and tuck it into the pot on top.
Add the rest of the shredded cabbage. In a small bowl combine the apple cider vinegar, water, brown sugar, salt and pepper; stir until the sugar dissolves, then pour this mixture evenly over the cabbage.
Dot the cabbage with the butter, cover the pot, and bring the liquid to a boil. Lower the heat to maintain a gentle simmer and cook, covered, for about 1½ to 2 hours until the cabbage is melt-in-your-mouth tender.
Remove and discard the clove-studded apple half. Give the cabbage a good stir to combine flavors, adjust seasoning if needed, and serve warm.