Season both sides of the chicken with ½ teaspoon each salt and pepper. Place the flour in a medium bowl. Dredge the chicken in the flour on both sides, shaking off the excess. Set the bowl of flour to the side.
Heat the butter and oil in a large cast-iron skillet over medium-high heat until the butter melts. Add the chicken to the pan and cook until browned, about 3 to 4 minutes on each side.
Remove the chicken from the pan, reserving the drippings.
Add the onion to the drippings and cook until softened, about 5 minutes. Add the garlic and cook 30 seconds more.
Stir together 3 tablespoons of the leftover flour, the poultry seasoning, paprika, and ½ teaspoon each salt and pepper. Sprinkle the flour mixture over the onions and garlic and stir. Gradually add the broth and milk, stirring until the flour dissolves in the liquid.
Nestle the chicken in the liquid and bring the mixture to a boil. Cover, reduce the heat to low and simmer for 20 minutes or until an instant read meat thermometer inserted in the thickest portion registers 165°F.
Sprinkle the chicken the parsley, if desired, and serve over the rice along with the pan gravy.