Into a blender or food processor add the first 9 ingredients: lemon juice, vinegar, garlic, pepper, salt, mayonnaise, Romano cheese, buttermilk, and sour cream. Pour the oil 3-4 tablespoons at a time, blending after each addition.
Tear Romaine lettuce into bite sized pieces. Add chopped chicken and dressing to lettuce. Assemble the pitas by adding the chicken and lettuce mixture to the insides and top with additional Romano cheese.