Pour apple juice into a large slow cooker. Mix in brown sugar and mustard, stirring until brown sugar has dissolved. Mix in the contents of the spice packets, then lay briskets into apple juice mixture. Top with potatoes, cabbage chunks, onions, and carrots, then push vegetables into liquid.

Cover and cook on High for 4 to 5 hours or on Low for 8 to 10 hours, until corned beef is very tender. Slice the meat thinly across the grain and serve with the vegetables.