1. Adjust oven rack to middle position and heat oven to 350°F. Heat 1 teaspoon oil in 12-inch skillet over medium-high heat until shimmering. Add sausage and cook, stirring to break sausage into 1/2-inch pieces, until browned, about 3 minutes. Transfer sausage to plate lined with paper towels. Remove skillet from heat; pour off fat into small bowl and reserve; wipe out skillet with paper towels.

2. Return skillet to medium-high heat and heat remaining 2 teaspoons oil until smoking. Pat chicken dry and liberally season with salt and pepper. Add chicken, skin side down, and cook without moving until well browned, 3 to 5 minutes. Using tongs, turn chicken and brown on other side, about 3 minutes. Transfer chicken to large plate. Remove skillet from heat and pour off fat into bowl with sausage fat; wipe out skillet with paper towels.

3. Return skillet to medium-high heat and heat 1 tablespoon reserved fat until shimmering. Add onion and cook until beginning to soften, about 2 minutes. Add bell pepper and cherry peppers and cook, stirring occasionally, until bell pepper begins to soften, about 4 minutes. Add garlic and cook, stirring frequently, until fragrant, about 30 seconds. Add sugar, 1/3 cup vinegar, and 3/4 cup broth; bring mixture to boil, scraping up browned bits from pan bottom.

4. Add sausage and chicken (with any accumulated juices) to skillet, arranging chicken pieces in single layer, skin side up, on top of peppers and onion. Transfer skillet to oven and cook until instant-read thermometer inserted into thickest part of chicken registers 160°F, 18 to 22 minutes, removing smaller pieces sooner if necessary. Meanwhile, combine cornstarch, thyme, and remaining tablespoon broth in small bowl.

5. Carefully remove skillet from oven (handle will be very hot) and transfer chicken, skin side up, to platter or individual serving plates. Place skillet over medium-high heat and stir in cornstarch mixture. Simmer sauce mixture until slightly thickened, 2 to 3 minutes. Off heat, taste sauce and add up to 2 tablespoons vinegar. Adjust seasonings with salt and pepper. Spoon sauce around chicken, being careful not to pour it directly over chicken. Sprinkle with parsley and serve immediately.

Serves 4 to 6