Heat the oil in a large saute pan over a medium heat. Add the onions and cook until they are soft and translucent, about 5 minutes. Add the garlic and cook for 1 minute more. Add tomato sauce, tomato paste, water, vinegar, molasses, pepper and liquid smoke and bring to a boil. Reduce heat to medium-low and simmer for 15 minutes. Add chopped chicken, return to a simmer, and cook an additional 10 minutes.
Split rolls. Place a leaf of lettuce on each roll, then pile on 3/4 cup of the chicken mixture onto the roll.