1. Heat olive oil and 1 tablespoon butter in a large heavy skillet set over medium heat. Season chicken with salt and pepper, rubbing it into the skin. Add chicken, skin-side down; cook, turning once, until well browned. Remove, and set aside.

2. Melt remaining 4 tablespoons butter in the same skillet. Sprinkle in flour, and cook, whisking constantly, until golden brown. Add onion, and cook until softened, about 2 minutes. Whisk in chicken broth; bring to a boil. Return chicken to skillet. Cook, partially covered, until chicken is cooked through and sauce has thickened. Transfer to a serving platter; garnish with parsley. Serve immediately.