Preheat the oven to 450 degrees F. Toss the broccoli with 1 tablespoon olive oil, 1/2 teaspoon salt and a few grinds of pepper on a baking sheet. Roast, stirring once, until tender and browned, 20 minutes.
Meanwhile, season the chicken with salt and pepper. Heat the remaining 2 tablespoons olive oil in a large skillet over medium-high heat. Add the chicken and cook, turning, until browned, about 8 minutes. Transfer to a plate.
Add the bell pepper, garlic and oregano to the skillet. Cook, stirring occasionally, until softened, about 3 minutes. Add the wine, scraping up any browned bits from the bottom of the skillet. Cook until reduced by half, 1 to 2 minutes. Add the cherry peppers and brine and the chicken broth. Nestle the chicken in the sauce and reduce the heat to medium. Cover and simmer until the chicken is cooked through, 5 to 8 minutes.
Remove the chicken to a cutting board. Increase the heat to medium high and continue simmering the sauce until slightly reduced, 2 to 3 minutes. Stir in the parsley and season with salt. Slice the chicken, divide among plates and top with the sauce. Serve with the broccoli.