Quick-Brined Corned Beef and Vegetables

In a large pot, bring 2 quarts water to a boil. Add kosher salt, pink curing salt, sugar, and spices; remove from heat and stir until both salts and sugar have dissolved. Let cool completely. Place brisket in a nonreactive container just large enough to hold it; pour cooled brine over meat. Place 2 small plates on top to keep meat submerged; cover and refrigerate 5 days.

Remove brisket; discard brine. Rinse brisket and place in a large pot. Add enough water to cover by 2 inches. Add halved onion, celery, and halved carrot; bring to a boil, then reduce heat to medium-low, cover, and simmer until very tender, 3 to 3 1/2 hours.

Meanwhile, set a steamer in a large saucepan. Add enough water to reach the bottom of steamer and bring to a boil. Add turnips, reduce heat to medium-low, cover, and steam until tender, 8 to 10 minutes. Transfer to a bowl. Repeat with small carrots and parsnips (together), steaming until tender, 10 to 12 minutes. Transfer to bowl.

Transfer beef to a cutting board. Tent with foil to keep warm. Strain broth through a fine-mesh sieve. Return all but 4 cups broth to pot; bring to a boil. Add cabbage, quartered onions, and parsley sprigs; simmer until very tender, about 35 minutes.

Meanwhile, in another pot, combine reserved 4 cups broth and potatoes. Bring to a boil, then reduce heat to medium and simmer until potatoes are tender, about 25 minutes. Strain (reserving broth), then toss potatoes with butter and chopped parsley; cover to keep warm. Add turnips, carrots, and parsnips to pot with cabbage mixture; cook until warmed through, about 10 minutes.

Remove and discard parsley sprigs; transfer vegetables to a platter with potatoes, reserving broth. Trim excess fat from beef. Slice thinly against grain, and transfer to platter. Serve with reserved broth and horseradish sauces.