Pat the corned beef dry with paper towels to ensure good browning. Heat the oil in a large skillet over medium-high heat until shimmering, then sear the meat for 2-4 minutes per side until deeply browned on the exterior. This creates a flavorful crust that adds depth to the final dish. Transfer the seared meat to a slow cooker. Pour the broth into the same skillet and add the mustard, scraping up all the browned bits from the bottom—these caramelized pieces are liquid gold for flavor. Once the mixture is boiling, pour it over the meat in the slow cooker.

Add the smashed garlic cloves, onion (cut in half), peppercorns, fresh thyme sprigs, bay leaves, and celery seed directly to the slow cooker with the meat and braising liquid. These aromatic ingredients will infuse the cooking liquid with complex flavors throughout the long braise. Cover the slow cooker and cook on low for 6-7 hours until the meat is very tender and easily pulls apart with a fork. I like to use freshly smashed garlic because it releases more natural oils compared to minced, giving a deeper, more robust flavor to the broth.

After the meat has cooked for 6-7 hours, carefully add the carrot chunks to the slow cooker around the meat, nestling them into the braising liquid. Continue cooking on low for another 1-2 hours until the carrots are fork-tender but still hold their shape. The vegetables will absorb the rich flavors from the broth while the meat continues to soften.

While the carrots are finishing in the slow cooker, melt the butter in a large skillet over medium heat. Add the cabbage wedges (keeping the core intact so they hold together) and season with salt. Sauté for about 10 minutes, stirring occasionally, until the cabbage begins to soften and develop light golden color on the edges. This preliminary cooking gives the cabbage a slight caramelization that adds sweetness and depth—it won't overcook when added to the slow cooker.

Transfer the sautéed cabbage wedges from Step 4 to the slow cooker, nestling them among the meat and carrots. Cook on low for 30-60 minutes more until the cabbage is tender and has absorbed the braising liquid's flavor. While this final braise happens, prepare or reheat your roasted potatoes and horseradish sauce according to their recipes. Once the cabbage is done, carefully remove the meat from the slow cooker and let it rest on a cutting board for 10 minutes—this allows the juices to redistribute through the meat, keeping it moist and tender when sliced.

Slice the rested meat against the grain into thick, even pieces—slicing against the grain breaks up the muscle fibers, making the meat more tender to bite. Arrange the sliced corned beef on a large platter or individual plates, then distribute the braised carrots, cabbage wedges, and roasted potatoes around it. Ladle some of the braising liquid from the slow cooker over the meat and vegetables. Garnish generously with finely chopped fresh parsley and fresh chives for a bright, fresh contrast to the rich, savory meat and vegetables. Serve with the horseradish sauce on the side.