Pour the apple juice into a large slow cooker, and mix in the brown sugar and mustard, stirring until the brown sugar has dissolved. Mix in the contents of the spice packets, and lay the briskets into the apple juice mixture. Top with red potatoes, carrots, onions, and cabbage chunks. Push all ingredients into the liquid.
Cover and cook on High until corned beef is very tender, 4 to 5 hours; or cook on Low for 8 to 10 hours. Slice meat thinly across the grain, and serve with vegetables.