Soak black beans in 3 cups water overnight. (Or place beans in a medium saucepan, and cover with 2 inches of water. Place lid on pan, and bring to a boil. Remove from heat, and let sit for 1 hour.) Drain beans in a colander, and set aside.
Set a small skillet over high heat. Place chiles in the pan; toast on both sides until browned and puffed, about 1 minute. Transfer chiles to a small bowl, and cover with 2/3 cup hot water. Let sit 10 minutes. Cut chiles in half, and remove the seeds and stems. Transfer the flesh, along with the poaching liquid, to the bowl of a food processor. Process until a puree forms.
Place a medium saucepan over medium heat. Add olive oil, and let warm. Add onions, 3/4 of the minced garlic, and cumin. Cover, and cook until softened and slightly browned, about 15 minutes. Add the chile puree and cook 2 minutes more. Add stock and the reserved black beans. Cover, and cook until beans are soft, about 1 1/2 hours. If beans seem dry, add another 1/2 cup hot water.
In a large bowl, toss together tomatoes, remaining minced garlic, basil, salt, pepper, and vinegar. Let sit 5 minutes. Stir half of the tomato mixture into the beans. Cover, and simmer 1 minute more.
Divide the chili among 6 bowls. Top with remaining tomato mixture, and serve.