Pat the brisket completely dry with paper towels and trim away any hard chunks of fat cap, leaving only a thin layer to keep it moist during the long cook.

Place the brisket fat-side up in the slow cooker insert.

In a small bowl, whisk together the barbecue sauce, beef broth, apple cider vinegar, and smoked paprika until the vinegar is fully dissolved.

Pour the sauce mixture evenly over the brisket, making sure to get it under the meat as well as over the top.

Cover the slow cooker and cook on low for 8 hours, or until the meat is fork-tender and easily pulls apart with a fork.

Remove the brisket to a cutting board, tent loosely with foil, and let it rest for 15 minutes before slicing.

While the brisket cooks, combine the green cabbage, red cabbage, cheddar cheese, and sliced apple in a large bowl.

In a small jar, shake together the mayonnaise, apple cider vinegar, sugar, and salt until the mixture is uniform and slightly thickened.

Pour the dressing over the cabbage mixture and toss thoroughly until every piece is evenly coated.

Slice the brisket against the grain into thin strips, then arrange on a serving platter alongside the slaw and spoon the remaining cooking liquid over the top for extra flavor.

Before serving, insert a food thermometer into the thickest part of the brisket to verify it has reached a safe internal temperature appropriate for your region, since slow cooker doneness can vary slightly based on appliance model and meat thickness.