In a small bowl, combine the oregano, 3/4 teaspoon salt, and 1/2 teaspoon pepper. Pat the chicken dry with paper towels, season with the spice mixture, and dredge in the flour.
Heat the oil in a Dutch oven over medium heat and brown the chicken, 4 minutes per side. Transfer to a plate.
Add the onions to the pot and cook until golden, about 5 minutes. Add the garlic and cherry peppers and cook, stirring, for 2 minutes. Add the pepper brine, wine, and broth and bring to a boil. Return the chicken to the pot and bring to a simmer. Cook, covered, until the chicken is cooked through, about 30 minutes.
Meanwhile, cook the rice according to the package directions. Divide among individual bowls and spoon the chicken and sauce over it.