Butterfly chicken breasts and set aside. In a food processor combine the dill and parsley and pulse until finely chopped. Reserve 1/4 cup of the herb mixture for spread, set aside.
In a plastic bag set in a medium bowl, combine remaining herb mixture, half the lemon juice, 1/4 cup olive oil, and salt and pepper, to taste. Place chicken in a plastic bag, add marinade, seal, and massage to distribute marinade evenly on the chicken. Refrigerate 1 hour, flipping once halfway through.
Meanwhile to make the herb spread, in a small bowl mix together remaining herb mixture, remaining lemon juice, lemon zest, sour cream, mayonnaise, pepperoncini, and salt and pepper, to taste.
Preheat grill to medium-high. Remove chicken from bag being careful not to remove herbs. Season with salt and pepper and cook on grill, turning once, until juices run clear, approximately 6 minutes on each side. Remove to a cutting board and cut each breast in half.
To assemble sandwich: cut ciabatta pieces in half horizontally, slather one side of bread with dill-pepperoncini spread. Place chicken on other slice, top with baby greens, and finish with the piece of bread covered with dill-pepperoncini sauce.