Place beans in a large, heavy pot. Add enough cold water to cover by at least 3 inches. Cover and soak overnight.
Place chicken in a large, heavy saucepan and add cold water to cover. Bring to a simmer. Cook until just tender, about 15 minutes. Drain and cool. Remove skin and cut chicken into cubes.
Drain beans into large colander. Heat oil in same pot. Add onions and saute over medium-low heat until translucent, about 10 minutes. Stir in garlic, green chiles, cumin, oregano, cloves and cayenne. Saute 2 minutes. Add beans and stock. Bring to a boil. Reduce heat and simmer until beans are very tender, stirring occasionally, about 3 to 4 hours. Add additional water if necessary. (Can be prepared 1 day ahead. Cover and bring to a simmer before continuing.) Add chicken, beer and 1 cup of cheese. Stir until cheese melts. Ladle chili into bowls. Serve with remaining cheese, sour cream, salsa and cilantro leaves to sprinkle on top.
To quick-soak beans, add enough water to cover by 3 inches. Bring to boil, cover and remove from heat. Soak for 2 hours.