Season the chicken with basil, salt and pepper to taste and place in a small heavy pot with just enough water just to cover (about 1 cup). Bring to boil then reduce to a simmer. Poaching chicken till almost cooked through about 10-15 minutes.
Remove the chicken and set aside to cool. Shred the chicken when cool enough to handle and discarding any fat.
Strain liquid and place liquid in a larger pot with the onion, pepper, garlic, barbecue sauce, vinegar and sriracha hot sauce, Worcestershire sauce and herbs and spices and bring to a boil.
Reduce the heat and add the shredded chicken, fresh basil and simmer for 30-60 minutes.
Spoon the mixture onto the rolls and top with the grated cheese and fresh greens.