Place a dry skillet over medium heat and toast the black peppercorns, mustard seeds, and coriander seeds until they become fragrant. Transfer the toasted spices to a mortar and pestle, and grind them into a coarse powder. Alternatively, you can use a spice grinder or leave the spices whole.
In a large pot or slow cooker, combine the ground spices (or whole spices), allspice berries, cloves, ginger, thyme, rosemary, bay leaves, and cinnamon sticks. Mix well to create the pickling spice blend.
Place the corned beef brisket in the pot or slow cooker, fat side up. Add the quartered onion, chopped carrots, celery, and minced garlic. Add sufficient water to the pot to ensure that the meat is completely covered.
If using a pot, bring the water to a boil, then lower the heat to a gentle simmer and let it cook for 3-4 hours, or until the corned beef reaches a tender consistency. If using a slow cooker, set it to low heat and cook for 8-10 hours, or until the meat is tender.
Once the corned beef is cooked and tender, remove it from the pot or slow cooker and let it rest for a few minutes. Slice the corned beef against the grain into thin slices.
Serve the corned beef slices with the pickling spice-infused broth and the cooked vegetables. You can also serve it with mustard or horseradish sauce for added flavor.