Preheat oven to 300. In a large Dutch oven, heat oil over medium high heat. Season chicken on all sides with salt and pepper. Sear, without crowding pot, until golden brown on all sides. Reserve on a platter.

Reduce heat to medium and add the onion, and garlic to the pot and cook until brown, about 7 minutes.

Deglaze Dutch oven with the beer and bring to a simmer. Then add salsas and chicken stock.

Return to a simmer and add the chicken and cilantro stems. Cover and place on the middle rack in oven. Cook until the meat is tender and falling off the bone, about 2 hours.

Remove chicken from the sauce and let cool until you can handle it. Remove the cilantro stems and discard.

Place Dutch oven over medium heat and simmer uncovered until the liquid is reduced by almost half. This should take about 20 minutes.

Remove from heat and stir in the cilantro leaves and lime juice. Adjust seasoning as necessary. Once the chicken is cool enough remove the skin and meat from the bone. Shred and place back in the sauce. Rewarm chicken in the sauce and serve.