Combine the chicken and enough water to cover in a large, heavy pot and bring to a boil over high heat.

Reduce the heat, cover the pot, and simmer gently until the chicken is cooked through, about 12 minutes.

Drain the chicken, reserving the cooking liquid.

Cool the chicken, remove and discard the bones, and coarsely shred the meat.

Meanwhile, place the same pot over medium-high heat and add the oil.

Add the onions and cook, stirring occasionally, until browned, about 10 minutes.

Add the garlic and cook 1 minute.

Stir in the chili powder, cumin, brown sugar, cocoa powder, oregano, and coriander and cook until the spices are fragrant, about 1 minute.

Add the tomatoes, beer, salt, and 3 1/2 cups of the reserved cooking liquid.

Bring to a simmer; reduce the heat to medium low, cover, and cook, stirring occasionally, 1 hour.

Add the beans. Simmer, uncovered, until the chili thickens, about 30 minutes.

Add the shredded chicken and simmer until heated through.