In a large stockpot, combine water, kosher salt, brown sugar, 2 tablespoons pickling spices, pink curing salt and garlic. Bring to a simmer, stirring until salt and sugar are dissolved. Remove from heat; cool to room temperature, then refrigerate until chilled.
Place 1 large oven roasting bag inside another. Place brisket inside inner bag; pour in cooled brine. Seal bags, pressing out as much air as possible; turn to coat meat. Refrigerate 10 days, turning occasionally to keep meat coated. Remove brisket from brine; rinse thoroughly. Place in a Dutch oven with water to cover. Add carrots, onions, celery and remaining pickling spices. Bring to a boil over high heat. Reduce heat; simmer, covered, adding water if necessary to keep brisket covered, until meat is tender, about 3 hours.
Serve warm or cool. Slice brisket thinly and serve in a sandwich or with additional vegetables simmered until tender in cooking liquid.
To make ahead: Refrigerate meat in cooking liquid for several days; reheat in liquid.