Remove the corned beef and spice packet from the packaging and set the spice packet aside. Give the corned beef a good rinse under cool water and place it, fat side down, in a large Dutch oven. Coat the top with half of the brown sugar. Flip the brisket over and coat the fatty side with the remaining brown sugar. Pour the Guinness around the meat adding just enough beef broth until the brisket is almost completely immersed. The fatty top of the brisket will be visible above the liquid. Reserve any remaining broth for later. Sprinkle the contents of the spice packet over the brisket.

Place the Dutch oven over HIGH heat and bring to a boil. Reduce to LOW, cover the pot, and simmer for 3 hours.

Remove the cooked brisket to a cutting board and tent with foil. Add the carrots and potatoes to the cooking liquid in the Dutch oven, adding enough remaining beef broth so that vegetables are almost covered. Increase the heat to HIGH to bring the liquid to a boil. Reduce the heat to MEDIUM-LOW, cover the pot, and simmer until the vegetables are fork tender, about 25 minutes.

Meanwhile, cut the cabbage into 5 or 6 sections. Transfer the cooked vegetables to a serving dish and place the sections of cabbage into the cooking liquid. Bring to a boil, then reduce the heat and simmer, covered, for about 5 minutes or until the cabbage has softened. Transfer the cabbage to a serving dish. Dot the vegetables and cabbage with a little butter and season with salt and pepper, to taste.

Slice off the fatty layer from the corned beef and slice it against the grain into 1-inch slices. Serve it with the cabbage, carrots, and potatoes, all drizzled with a little of the cooking liquid.