1. Preheat oven to 400°F. Rinse chicken pieces with cool water and pat dry. Season on all sides with salt and pepper.
2. In a Dutch oven, melt butter with oil over medium heat. When foam subsides, cook chicken, working in batches if necessary, turning often, until golden, about 10 minutes. Transfer to a plate. Pour off all but 1 Tbsp. fat. Add vegetables; cook, stirring often, until slightly softened, about 2 minutes. Pour in wine and cook, scraping up any browned bits, until liquid has almost evaporated, 30 seconds to 1 minute. Add broth, garlic and rosemary. Season with salt and pepper and bring to a boil.
3. Return chicken, skin side up, and any accumulated juices to pot. Cover and bake for 20 minutes. Remove cover; bake until juices have slightly reduced and garlic is very soft, about 30 minutes. (If mixture looks dry, add up to 1/2 cup broth.)
4. Transfer chicken to a serving platter. Use a fork to crush a few of the softened garlic cloves. Spoon sauce and vegetables over chicken and serve immediately.