Pat chicken pieces dry with paper towels and season with 2 teaspoons of salt and 1/2 teaspoon ground pepper. Place the flour in a large shallow dish. Dredge the chicken pieces in flour, shaking off excess, and set aside. Save the flour to use later in the recipe.
Heat the butter and oil in a large cast iron pan or a saute pan over medium-high heat. Working in two batches, place the chicken pieces skin side down on the pan and cook for about 4-5 minutes, until deep golden brown. Flip and cook on the other side for another 4-5 minutes. Adjust the heat if the chicken is browning too fast. Set the chicken pieces aside.
In the same pan, saute the onions and celery, seasoned with a pinch of salt and pepper, over medium-high heat until softened, about 5 minutes. Add the garlic, sage and 2 tablespoons of flour left over from dredging. Stir well to ensure that the flour is nicely incorporated. Add the broth and the vinegar, then scrape and stir the browned bits from the bottom of the pan to incorporate all cooking residues into the liquid.
Transfer the chicken into the pan and nestle into the sauce. Simmer over low heat for about 35 to 40 minutes, or until the chicken drumsticks and thighs reach 185F internal temperature.
Transfer the chicken to a serving platter. Taste the sauce and adjust for salt and pepper. Spoon the sauce over chicken and sprinkle with fresh parsley before serving.