Set the corned beef in a large pot with the onions, celery, and pickling spices, then pour in water until its surface sits 2 inches above the meat and season with salt and pepper. The brisket should sit beneath the water with room around it for the aromatics.

Heat the pot until the water boils, then turn the heat to low, cover, and cook at a gentle simmer for 3 to 3 1/2 hours. A fork should enter the beef with little resistance when it becomes very tender.

During the braise, whisk the Dijon mustard with the apple cider vinegar in a small bowl and season with salt and pepper. The finished sauce should look uniform and have a sharp, tangy flavor.

Stir the mayonnaise, chopped capers, caper brine, and parsley together in another bowl, then season with salt and pepper. The mixture should remain creamy, with capers and parsley distributed throughout.

Lower the cabbage and carrots into the pot and keep simmering for another 45 minutes to 1 hour, until the cabbage is tender. The cabbage leaves should soften and become more translucent while the wedges retain enough structure to lift out.

Transfer the beef, cabbage, and carrots out of the broth, then carve the meat across the grain and finish with salt and pepper to taste. The slices should hold together while yielding easily when cut with a fork.

Set both dipping sauces beside the beef and vegetables for serving. The mustard sauce will be sharp and smooth, while the mayonnaise sauce will be creamy and flecked with herbs and capers.