For the brine, place all of the brine ingredients into a large saucepan and cover with 5 litres/8¾ pints of water. Bring the mixture to the boil and cook for 2-3 minutes. Remove the pan from the heat and set aside to cool completely.
Place the beef into a large non-metallic container and pour over the cooled brine, (ensure the beef is submerged in the brine). Set aside somewhere cool, preferably the fridge, for 3-5 days. After the soaking time in the brine, rinse the beef brisket and soak in cold water for 24 hours.
Rinse the beef brisket and place into a large saucepan with the vegetables. Cover the brisket completely with cold water and bring the mixture to the boil. Reduce the heat and simmer for 2-2½ hours, or until the beef is very tender. Drain well.
Meanwhile for the horseradish mash, boil the potatoes in a large pan of salted water, for 10-15 minutes, or until tender. Drain well. Return the potatoes to the pan and heat over a low heat for 10-15 seconds to drive off any excess moisture.
Mash the potatoes until smooth using a potato masher or ricer, then beat in the butter. Stir in the horseradish and chives and season, to taste, with salt and freshly ground black pepper.
To serve, carve the beef into thick slices and serve with the horseradish mash, vegetables and gherkins.