Brussels Sprouts with White Beans and Pecorino

Heat 3 tablespoons oil in heavy large skillet over high heat. When just about to smoke, add half of brussels sprouts. Cook until brown, stirring occasionally, about 5 minutes. Transfer to large bowl. Heat 3 tablespoons oil in same skillet. Add remaining brussels sprouts, reduce heat to medium-high, and cook until brown, stirring occasionally, about 5 minutes. Transfer brussels sprouts to same bowl. Add remaining 2 tablespoons oil to skillet; increase heat to high. Add garlic; sauté until brown, stirring constantly, about 1 minute. Add broth and brussels sprouts. Cook until brussels sprouts are crisp-tender, stirring frequently, about 3 minutes. Add beans and butter; stir until butter melts and broth is reduced to glaze, about 1 minute. Season with salt and pepper. Stir in cheese.