Pick over and rinse the beans. Put them into a large bowl and cover completely with cold water. Let the beans soak overnight (or at least 8 hours).
Drain the beans and transfer them to a medium saucepan. Add the 2 cups water and the bay leaf. Bring to a boil over medium heat and cook for 15 minutes. Reduce the heat to low and simmer, uncovered, for about 20 minutes, until the beans are tender. Drain the beans and discard the bay leaf.
Preheat oven to 400ºF.
Place a heavy stockpot over medium heat for about 1 minute, then spray it twice with the vegetable oil. Add the broccoli stems, carrot, celery, and onion. Cover, reduce the heat to low, and cook for 5 minutes, stirring once or twice. Stir in the thyme, basil, and wine. Simmer, uncovered, for about 15 minutes, until the wine has been reduced by 1/2.
In the meantime, coat the chicken thoroughly with the barbecue sauce and bake for 10 minutes on the top shelf of the oven. Remove the chicken from the oven and allow it to cool just long enough to handle. Cut the chicken into small cubes.
Add the chicken, chicken stock, and beans to the stockpot. Cook over low heat for about 3 minutes, until thoroughly heated. Stir in the evaporated milk and the broccoli florets. Cook for 5 minutes, stirring if needed to keep the soup from coming to a boil. Add the dissolved cornstarch and cook for 2 minutes more, stirring constantly. Stir in the liquid smoke, worcestershire, and tabasco sauce.