Preheat the broiler. Put the potato in a medium saucepan, cover with water and add 1 tablespoon salt. Bring to a simmer and cook until just tender, about 8 minutes. Drain well. Meanwhile, pat the chicken dry and sprinkle with the oregano, and salt and pepper to taste. Heat 2 tablespoons olive oil in a large cast-iron skillet over high heat. Add the chicken and cook until browned, about 5 minutes per side. Transfer the chicken to a plate. Add the potato to the skillet and cook, stirring, 3 minutes. Add the onion and bell pepper and cook until slightly soft, about 1 minute. Reduce the heat to medium; add the remaining 1 tablespoon olive oil and the garlic and cook 2 minutes. Add the cherry peppers and brine and cook, scraping up the pan with a wooden spoon, about 1 minute. Remove from the heat. Return the chicken to the skillet and sprinkle with the parsley. Toss with tongs, then set the chicken skin-side up on top of the vegetables. Broil until the skin is crisp, 2 to 3 minutes.