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Preheat the oven to 190C/Gas 5/fan oven 170C. Wash and dry brisket and season. Heat 2 tablespoons of oil in a deep casserole and brown beef all over. Remove from pan. Turn down heat, add butter and fry the onions, celery, carrots and mushroom stalks for 6-8 minutes.
Return beef to pan and add beer, thyme, bay leaves and sugar. Add water if necessary so the liquid comes about two-thirds up the beef. Season, bring to a simmer, cover tightly, and cook in the oven for 20 minutes. Reduce heat to 160C/Gas 3/fan oven 140C and cook for 2 hours, turning twice, until tender.
An hour before the beef is done, toss the parsnips in oil, season and roast on a baking tray above the beef for 50 mins - 1 hr until tender, turning once.
Turn oven up to 190C/Gas 5/fan oven 170C. Lift out the beef, tent with foil and keep warm. Stir the parsnips and mushroom caps into the beef juices. Check seasoning; add water if needed. Cover and cook in the oven for 20-25 minutes until mushrooms are tender.
To serve, use a slotted spoon to remove vegetables and arrange round the beef. Spoon off the excess fat from the juices, then whisk in the mustard and pour into a jug. Moisten the beef with a little juice and scatter with parsley or thyme. Serve with mash, if you like.