1. Put the kosher salt, sugar, garlic, thyme, pickling spice, and 1 quart water in a 6- to 8-quart pot. Bring to a boil over high heat, sitrring to dissolve the salt and sugar. Remove from the heat and let the mixture steep for 5 minutes. Stir in the curing salt to dissolve, add 3 quarts of cool water, and transfer to the refrigerator to chill for at least 2 hours.
2. Put the brisket in a lidded, high-sided, food-grade plastic or metal container just large enough to contain the meat. Pour in the cooled curing liquid to cover the meat, weighting it with small plates, if necessary, to keep the beef submerged beneath the brine. Cover and refrigerate for 5 days, turning the meat over after 2 or 3 days.
3. Remove the meat from the liquid and place in a pot large enough to just contain it. (Reserve the curing liquid for now.) Rinse the meat in two or three changes of water and drain. Strain the herbs, spices, and garlic from the curing liquid and discard the liquid. Add the spices to the pot with the meat and cover by 2 inches of water. Add the onion, celery, and carrot and bring just to a simmer over medium-high heat. Cover, reduce the heat to maintain the barest possible simmer, and cook until very tender but not yet falling apart (a carving fork should slide easily into the meat), 3 to 4 hours.
4. Carefully transfer the corned beef to a cutting board and, using a sharp knife, slice thinly across the grain. Serve each potion moistened with a few tablespoons of the cooking liquid (reserve enough cooking liquid to reheat any remaining beef).