Heat 2 teaspoons oil in a large nonstick skillet over medium-high heat. Sprinkle 1/4 teaspoon salt over chicken. Add chicken to pan; cook 1 minute on each side or until lightly browned. Add 1/4 cup vinegar; cook 2 minutes or until chicken is done and vinegar is syrupy. Remove chicken mixture from the pan; cover and keep warm. Wipe pan clean with a paper towel.
Return pan to medium-high heat; add remaining 2 teaspoons oil. Add bell pepper and onion; sauté 7 minutes or until tender. Stir in remaining 1/4 teaspoon salt and remaining 1/4 cup vinegar; cook 1 minute or until vinegar is syrupy.
Arrange chicken mixture evenly over bottom halves of bread; top with bell pepper mixture. Arrange cheese over pepper mixture, and sprinkle with black pepper. Top with top halves of bread. Place a cast-iron or heavy skillet on top of sandwiches; let stand 5 minutes. Cut each sandwich into 6 wedges.