Heat the oil a large, heavy bottomed Dutch oven over medium-high heat until beginning to smoke. Season the brisket generously with the paprika, salt, and pepper. Place fat side down in the Dutch oven and cook until brown on all sides, about 8 to10 minutes. Transfer the meat to a plate and set aside.
Discard about half the fat, then add the onions and cook over medium heat until browned and tender, about 10 minutes. Pour in the vermouth and cook until reduced by about half. Nestle the brisket into the onions fat side up, add the broth; bring to a simmer and bake, covered, until the brisket is fork tender, about 1 1/2 to 2 hours. Uncover, scatter the carrots around the brisket and cook until the carrots are tender, about 30 minutes more.
Transfer the brisket to a cutting board, cut the fat off the top and thinly slice the meat across the grain. Return the meat to the braising liquid and serve immediately or, cool to room temperature in the sauce, cover and refrigerate overnight. When ready to serve reheat in the sauce.
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