Rinse the brisket all over, then pat very dry. If applicable, trim off excess fat carefully with a paring knife, being careful not to cut away any of the meat itself.
Place the brined brisket into a large stockpot (6 quarts or larger) and cover with at least 1 inch of water above the brisket. Add garlic cloves, bay leaf, and 1 tablespoon pickling spice.
Bring liquid to a boil over high heat, then reduce the heat to maintain a low simmer. Cover and let simmer for about 3 hours or until the brisket registers 180 degrees F to 190 degrees F on a meat thermometer and is fork tender.
Once the meat is ready, turn off the heat and let cool 10 minutes. With kitchen tongs, carefully remove the corned beef from the cooking liquid and transfer it to a cutting board. Remove 2 cups of liquid from the pot, then strain it into a bowl to remove any pieces. Set aside. Discard the remaining liquid from the pot and lightly rinse the pot to use later.
Let the corned beef cool for 20 minutes at room temperature.
While the corned beef cools, place the 2 cups of the strained liquid and 2 cups of water into the now-empty pot.
Top with carrots, potatoes, onion, and cabbage. Bring liquid to a boil, then use a soup ladle to baste the vegetables at the top of the pot. Reduce heat to a low simmer, cover, and cook until vegetables have softened, about 25 to 30 minutes. Baste the vegetables with the hot liquid 2 more times as they cook.
While the vegetables cook, slice the corned beef against the grain into 1/4” inch strips.
When the vegetables are ready, use a slotted spoon to transfer them to a large serving platter that has a lip to prevent liquid overflow. Top the cooked vegetables with the sliced corned beef.
To the pot of leftover liquid from the vegetables, add the mustard and mix together until combined. Spoon the liquid over the platter. Enjoy!
*Depending upon your brand of pickling spice, it can be a little heavy on the allspice. If you prefer, you can pick out a few of the allspice berries, or enjoy it as is.
TO USE STORE-BOUGHT CORNED BEEF: Rinse a 3- to 4-pound corned beef brisket and pat before cooking as directed. If the corned beef comes with a pickling spice packet (sometimes the store-bought versions do), you can use this in place of the “1 tablespoon pickling spice” when cooking the corned beef, so you don’t need to buy a jar of pickling spice.
TO STORE: Refrigerate corned beef and cabbage in an airtight storage container for up to 4 days.
TO REHEAT: Gently rewarm leftovers in a covered baking dish in the oven at 350 degrees F or in the microwave.