Heat oil in a large Dutch oven over medium-high heat. Add the chopped onion and cook, stirring often, until onion softens and just begins to brown. Add beef broth, corned beef and the contents of the spice packet, and enough water to cover by about 1 inch; bring to a boil. Reduce head to a medium simmer, cover, and cook, stirring occasionally, for 1 hour per pound.

About 30 minutes before brisket is finished, core cabbage, discarding core, cut into large chunks, and add to pot. Peel and trim onion; halve vertically and cut into several wedges and add to pot. Add additional water if needed to cover. Return to a simmer and cook, undisturbed, for another 45 minutes, or until corned beef is quite tender. Add carrots and potatoes, bring to a high simmer, and cook until vegetables are tender, about 15 minutes.

Remove from heat, and remove corned beef to a wire rack set over a rimmed baking sheet; allow to cool and drain for 10 minutes. Transfer to a platter, returning any accumulated juices to the pot. Slice corned beef thinly across the grain and serve in soup bowls with some of the cooking liquid and a selection of the vegetables. Have mustard, horseradish, and bread on the table.