How to Make It
Preheat oven to 350°F. Heat 2 tablespoons canola oil in a large Dutch oven over high. Sprinkle lamb evenly on all sides with salt. Place lamb, fat side down, in hot oil; cook, undisturbed, until golden brown, about 12 minutes. Remove from heat. Turn lamb fat side up; drizzle with remaining 1 tablespoon oil. Add rosemary, star anise, and garlic to Dutch oven. Cover tightly, and transfer to preheated oven. Cook until lamb is falling off the bone, 3 hours and 30 minutes to 4 hours. Remove from oven; tent with aluminum foil, and let lamb stand 15 minutes. Tear lamb meat into chunks or shreds, and discard bone. Cover to keep warm, and set aside.
Bring broth to a boil in a medium saucepan over high. Stir in piment d’Espelette and salt; remove from heat. Gradually add polenta, whisking constantly. Cover and let stand 45 minutes. Add cheese and butter, whisking vigorously to break up any lumps. Keep warm.
Bring a large pot of heavily salted water to a boil over high. Fill a large bowl with ice water. Add sprouts to boiling water; cook, stirring once, until bright green and just tender, 3 to 5 minutes. Using a slotted spoon, transfer sprouts to ice bath; stir to stop the cooking process. Drain and pat dry. Melt butter in a large skillet over medium-high. Add sprouts; cook, stirring occasionally, until golden brown, about 10 minutes.
To serve, reheat polenta over low, stirring often, until warmed through. Spoon polenta into 4 bowls; top with brussels sprouts and lamb.