Preheat oven to 375°F (190°C). Pat the chicken pieces dry with paper towels and season all over with salt and black pepper.
In a large skillet over medium heat, melt 2 tablespoons of butter. Brown the chicken pieces, skin side down, for 3 to 4 minutes per side until lightly golden. Transfer browned pieces to a large baking dish.
In the same skillet, add the remaining butter and sauté the sliced onion until soft, about 4 minutes. Add the minced garlic and cook 30 seconds more.
Stir in the cream of chicken soup and chicken broth, scraping up any browned bits from the pan. Whisk in the flour until smooth and simmer 2 minutes to thicken slightly.
Pour the sauce evenly over the chicken in the baking dish, covering pieces as much as possible. Cover the dish tightly with foil.
Bake covered for 60 minutes. Remove foil and bake an additional 25 to 30 minutes until the chicken is cooked through (internal temperature 165°F) and the gravy is bubbling and slightly thickened.
Let the chicken rest 10 minutes before serving. Sprinkle with chopped parsley and serve the chicken with the pan gravy spooned over the top.
Notes
Leftovers keep well refrigerated for up to 3 days. Reheat gently to avoid drying the chicken; add a splash of broth if gravy thickens too much.