Peel the carrots and cut them into 2-inch chunks. Cut the cabbage into 8 wedges, keeping the core intact so the wedges hold together during cooking. Pat the corned beef dry with paper towels—this helps it brown better if you decide to sear it first, though it's optional for a slow cooker. Arrange the carrots and corned beef in the bottom of your slow cooker, then nestle the cabbage wedges around them.

In a bowl, whisk together the apple juice, brown sugar, and 1 tbsp of mustard until the sugar dissolves. This combination creates a slightly sweet and savory braising liquid that will infuse the corned beef and vegetables with flavor. I like to taste it quickly—the apple juice should be the dominant flavor, with subtle sweetness and mustard background.

Pour the braising liquid from Step 2 over the corned beef and vegetables in the slow cooker. The liquid should come about three-quarters of the way up the meat—it doesn't need to cover everything completely. Cover and cook on low for 10-12 hours or on high for 5-7 hours, depending on your schedule. The meat is done when it's very tender and pulls apart easily with a fork.

Remove the corned beef from the slow cooker and let it rest for 5 minutes before slicing against the grain—this keeps it from falling apart and ensures tender, clean slices. I always slice thicker pieces (about 1/4 inch) rather than thin ones because the corned beef is so tender it needs the structure. Arrange the sliced beef on a platter with the cooked carrots and cabbage wedges from the slow cooker.

Serve the corned beef, carrots, and cabbage with a ramekin of the remaining 4 tbsp mustard on the side for individual taste preferences. Place a dollop of softened butter on each serving or pass it separately—the butter adds richness that balances the mustard's sharpness. The braising liquid can be strained and served as a light sauce if desired, though most people enjoy the meat and vegetables plain with just mustard and butter.