Drain the soaked salt beef and place it in a large, heavy-bottomed stockpot, covering it completely with fresh cold water. Bring the liquid to a boil over high heat, then reduce the heat to medium-low, cover, and simmer gently for 60 minutes until the beef begins to tenderize.

While the beef is simmering, place the rinsed yellow split peas into a clean cotton pudding bag or double-layered cheesecloth pouch, tying it securely with kitchen twine but leaving some room for the peas to expand. Submerge the bag of peas directly into the boiling stockpot alongside the beef.

After the beef and peas have simmered together for 60 minutes, carefully add the turnip wedges and carrots to the pot. Cover and continue to simmer for an additional 20 minutes, allowing the root vegetables to absorb the rich, salty broth.

Add the whole peeled potatoes and cabbage wedges directly on top of the other ingredients in the pot. Cover tightly and cook for 30 to 40 minutes, or until the potatoes are fork-tender and the cabbage is beautifully soft and translucent.

Carefully transfer the boiled vegetables and salt beef to a large serving platter using a slotted spoon, keeping them warm. Lift the split peas bag out of the pot, open it carefully, and empty the cooked peas into a medium bowl.

Mash the hot split peas with the butter and black pepper until a thick, smooth pease pudding forms. Arrange the sliced salt beef in the center of your platter surrounded by the vegetables, and serve immediately with the pease pudding and pickled beets on the side.