Whisk together the barbecue sauce, one tablespoon of mustard, and a ½ teaspoon salt. Rub barbecue mixture over pork, and place pork in a slow cooker. Pour water over top, and cook on low for 7 hours until very tender.
Remove pork from slow cooker, reserving liquid for later use, and shred using two forks.
Meanwhile, pour the reserved barbecue liquid into a large sauce pan. Stir in the extra virgin olive oil, ½ tablespoon mustard, vinegar, ½ teaspoon salt, and caraway seeds. Bring to a boil.
Add the chopped cabbage, and simmer gently until just wilted about 3 to 4 minutes. Divide the cabbage slaw into 4 equal-sized portions and serve each with 6 ounces shredded pork.