In a smoker, using hickory chips, smoke the chicken breast and 1 head of the garlic for 2 to 3 hours. If you do not have a smoker, substitute already-smoked chicken and roast 1 head of garlic in the oven. To roast the garlic, peel off the outer skin, leaving the cloves and head intact. Place the head on a double thickness of aluminum foil and brush with 1 teaspoon olive oil. Seal the foil and bake at 375 degrees F for 45 to 55 minutes.
Remove the cloves from the other head of garlic. Finely chop 8 cloves and set aside. Then finely chop 3 more cloves (or to taste) and set aside. Heat the oil and butter in a cast-iron skillet. Add the 8 chopped garlic cloves and red onion and saute until golden. Shred the smoked chicken and add to the skillet. If you purchased smoked chicken rather than cooking your own with garlic, add the roasted garlic to the chicken-onion mixture by squeezing the garlic from the inner skins. Discard the skins. Add the tomato, thyme, white pepper, and salt and black pepper to taste. Mix well. Add the wine and stock and cook over medium heat for 5 minutes. Add the cream and cook over low heat for 8 minutes more. Add the remaining 3 chopped garlic cloves. Pour over the pasta and garnish with the parsley.