Place chicken between 2 sheets of heavy-duty plastic wrap, and flatten to 1/4-inch thickness using a meat mallet or rolling pin.
Stir together salsa and next 3 ingredients. Remove 1 cup mixture, and set aside remaining mixture.
Place chicken in a shallow dish or heavy-duty zip-top plastic bag; pour 1 cup salsa mixture over chicken. Cover or seal, and chill 1 hour, turning occasionally. Remove chicken from marinade, discarding marinade.
Grill peppers, covered with grill lid, over medium-high heat (350° to 400°) 5 to 7 minutes, turning often, until peppers look blistered.
Place peppers in a heavy-duty zip-top plastic bag; seal and let stand 10 minutes to loosen skins. Peel peppers; remove and discard seeds. Cut peppers into thin strips; set aside.
Grill chicken, covered with grill lid, over medium-high heat (350° to 400°) about 5 minutes on each side or until done. Cool slightly. Cut chicken into thin slices, and sprinkle evenly with salt.
Stir together beans and olive oil in a 1-quart glass bowl, and microwave at HIGH 2 minutes or until thoroughly heated, stirring once.
Spread beans evenly over bottom halves of rolls. Spread avocado over top halves of rolls. Top bottom halves evenly with chicken, pepper strips, cheese, and top halves of rolls. Serve with reserved salsa mixture.