Start the corned beef with a fresh brisket about 7 days before you plan to make your dinner.
Combine water, salt, sugar, mustard, spices, and 3 cloves minced garlic in an 8-quart nonreactive stock pot. Bring to a boil over high heat.
Remove from heat and set aside to cool. When liquids have cooled, trim the corned beef of excess fat and put into the liquid.
You may need to transfer the liquids and corned beef to a larger container or bowl, just make sure it is nonreactive. If necessary, add more cold water to cover the beef. Using a heavy pottery bowl or a few small bowls, weigh the corned beef down so it will stay submerged in the water.
Cover the pan and refrigerate for 5 to 7 days in the refrigerator. Take out once a day to stir lightly, and turn the beef as needed.
On the 6th or 7th day, remove the beef from the brine and rinse well with cold water. Place in a large (8-quart or larger) pot. Cover with cold water and add the celery, onions, and 2 cloves minced garlic. Bring to a boil; skim off any scum which develops on the surface. Reduce heat to medium-low; cover and simmer for 2 1/2 hours.
Uncover the pot and add the potatoes, carrots, rutabaga or turnip, and cabbage. Cover the pot and cook until meat and vegetables are tender, about 1 hour longer. If cooking some vegetables in another pot, cook just until tender and keep warm.