1. Preheat oven to 325 degrees with rack in lower third. Season brisket with 1 tablespoon salt and 1/2 teaspoon pepper. Heat 2 teaspoons oil in a dutch oven or other ovenproof heavy large pot over medium-high heat. Add brisket, fat side down, and cook, flipping once, until well browned on both sides, about 10 minutes.

2. Transfer brisket to a plate; lower heat to medium. Saute shallots, stirring frequently, until golden, about 8 minutes. Add pomegranate juice, increase heat, and boil until reduced by half, about 5 minutes. Return brisket, fat side up, and any accumulated juices to pot. Add enough broth to come three-quarters up brisket. Cover and transfer to oven.

3. Braise brisket, spooning juices over occasionally, until fork-tender, 3 to 3 1/2 hours. Uncover and let cool slightly. Shred in pot with 2 forks, tossing until meat has absorbed most of braising liquid. Cover to keep warm.

4. Increase oven to 400 degrees with racks in upper and lower thirds. Toss brussels sprouts and mushrooms with 2 tablespoons oil and 1 teaspoon salt; transfer to a rimmed baking sheet. Toss fennel with remaining tablespoon oil and 1/2 teaspoon salt; transfer to another rimmed baking sheet. Roast sprouts and mushrooms 10 minutes. Add fennel to oven and continue roasting, switching rack positions halfway through, until vegetables are tender and golden brown, 30 to 35 minutes more. Serve vegetables warm or at room temperature with brisket, rolls, cheese, dates, and condiments. Brisket can be made up to 2 days ahead. Uncover and let cool completely in braising liquid, but do not shred. Cover and store in braising liquid in refrigerator. Reheat, covered, over medium-low heat until warmed through; then shred.