Preheat oven to 325°F. Heat oil in a Dutch oven over medium-high heat. Add brisket and cook until browned, 3 to 5 minutes per side. Transfer to a large plate and set aside.
Add onions to the pot; cook, stirring frequently, until softened, about 2 minutes. Stir in allspice, thyme, paprika, salt, pepper and bay leaves, then pour in vermouth (or wine). Bring to a boil. Cook for 3 minutes.
Stir in broth and return the brisket to the pot along with any accumulated juices. Bring to a simmer. Cover, place in the oven and bake for 1 1/2 hours. Meanwhile, cut carrots, parsnips and rutabaga into 2-by-1/2-inch sticks.
Transfer brisket to a plate. Using a slotted spoon, remove and discard bay leaves and allspice berries (if using). Stir mustard into the sauce. Add the carrots, parsnips and rutabaga. Return the brisket to the pot; cover and bake for 1 hour more.
Test vegetables and brisket for tenderness by piercing with the tip of a sharp knife. As they get done, transfer to a cutting board or platter, cover with foil and set aside. If necessary, continue to cook, testing for doneness every 20 minutes. Total cooking time for the brisket may range from 2 1/2 to 5 hours, depending on the particular piece of meat.
Skim fat from the sauce. Place the pot over high heat and bring to a boil. Cook for 5 minutes, stirring occasionally, to reduce and intensify flavors. Dissolve arrowroot in 1 tablespoon water (or cornstarch in 2 tablespoons water); add to the simmering sauce and cook, stirring constantly, just until thickened, about 10 seconds.
Slice the brisket thinly against the grain and arrange slices on a serving platter. Using a slotted spoon, mound the vegetables around the brisket. Spoon half the sauce over the meat and vegetables; pass remaining sauce separately.
Ingredient note: Brisket cuts are notoriously fatty. But the flat, first-cut section is a far better choice for healthy eating than the fattier point cut. Don't worry about a first-cut being tough there's enough juice in this melange of root vegetables to keep the meat moist, no matter how lean it is.
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