Melt the butter in a large skillet over medium-high heat. While it heats, season the chicken pieces with 1 teaspoon of salt and ½ teaspoon of black pepper.

Coat the seasoned chicken in the ¾ cup flour placed in a shallow dish, ensuring each piece is well dredged. Brown the floured chicken in the hot butter on all sides until a rich golden crust forms. Once browned, remove the chicken and set aside. Drain the skillet, reserving approximately 1 tablespoon of the butter.

Lower the heat to medium-low. Add the chopped onions, carrots, celery, and garlic to the skillet. Sauté the vegetables, stirring occasionally, until softened, about 5 minutes.

Sprinkle in the 3 tablespoons of flour, mixing thoroughly with the vegetables. Allow this mixture to cook for an additional 5 minutes, letting the flour toast slightly.

Gradually pour in the chicken broth, stirring constantly to create a smooth base. Season with the remaining salt, remaining black pepper, and cayenne pepper. Bring the sauce to a boil, then immediately lower the heat to maintain a gentle simmer.

Return the browned chicken pieces to the skillet, nestling them into the gravy. Cover the skillet with a lid and allow the dish to cook for about 30 minutes, or until the chicken is thoroughly cooked and the gravy has thickened to a rich consistency.