Combine first 6 ingredients, and sprinkle 4 teaspoons oregano mixture evenly over chicken. Set remaining oregano mixture aside.
Heat oil in a large Dutch oven over medium-high heat. Add the chicken to pan; cook 2 minutes on each side or until browned. Remove chicken from the pan. Add mushrooms, bell peppers, and onion to pan; sprinkle with remaining oregano mixture. Cover and cook for 5 minutes or until tender, stirring occasionally. Add vinegar, bay leaves, tomatoes, and tomato paste; stir to combine. Arrange the chicken over tomato mixture, and bring to a boil. Cover, reduce heat, and simmer 50 minutes or until the chicken is done.
Place 1 cup pasta in each of 6 shallow bowls; top each serving with 1 chicken breast half. Ladle 1 cup sauce over chicken, and sprinkle each serving with 4 teaspoons of Parmesan cheese.