Spear the peppers with a kitchen fork or skewer and hold them over an open flame until charred all over. Place the charred peppers in a paper bag or in a covered container until they are cool enough to handle. (This step completes the cooking of the peppers and facilitates removing the skin.) Rub off the skins (never run them under water, which washes away the flavorful oils), cut the peppers in half lengthwise, and discard the stems and seeds. Cut the pepper halves into 1/4-inch wide strips and set aside.
Heat the oil in a large skillet over medium-high heat. Sprinkle the chicken pieces with 1 1/2 teaspoons salt and 1/2 teaspoon pepper and cook until lightly browned, about 2 minutes on each side. Transfer the chicken to a plate and reserve.
Add the sausage to the skillet and cook until browned, about 2 minutes. (Reduce the heat as needed to keep the skillet from burning.) Toss in the mushrooms and cook along with the sausage, stirring until softened, about 2 minutes. Add the peppers and stir in the flour. Cook for 1 minute, stir in the vinegar, broth and tomato sauce and bring to a simmer, scraping the bottom of the pan to deglaze. Season with 1/2 teaspoon salt and a pinch of pepper.
Return the sauteed chicken pieces to the skillet along with any juices that have accumulated on the plate. Cover the skillet and simmer gently, until the chicken is just cooked through, 30 to 35 minutes. Transfer the cooked chicken to a warm serving platter. Return the skillet to the heat and reduce the liquid until it thickens to a sauce consistency, about 1 to 2 minutes. Taste and adjust the seasoning. Spoon the sauce, mushrooms, peppers and sausage over the chicken. Sprinkle with the chopped parsley, and serve immediately.
Recipe courtesy of Michael Romano