Combine the chili powder, cayenne, and 1/2 teaspoon salt in a small bowl. Season the chicken thighs on both sides with the spice blend.
Add the chicken to the pot in batches, turning the heat down to medium-high if smoking, until nicely browned on both sides, about 2 minutes per side. Remove to the bowl with the bacon.
Add the onions to the pan and turn the heat down to medium. Season with salt and saute, scrapping up brown bits from the bottom of the pot, until soft, about 5 minutes. Add the garlic and saute for another minute. Pour in the chicken broth, white beans, green chiles, and cilantro stems. Fold in the reserved chicken and bacon. Bring to a simmer, then cover with the lid, turn the heat down to low, and cook for one hour, until the chicken is tender, but not falling apart.
Remove the chicken to a bowl and shred using two forks. Puree 2 cups of the chili using a blender, or simply use an immersion blender and puree about 1/4 of the mixture. Return the pureed beans to the pot along with the shredded chicken. Place the lid back on the pot, and simmer on low for another 30 minutes, or up to 2 hours, until the broth has thickened and the chicken is finely shredded.
Garnish with the cilantro leaves and serve alongside shredded cheddar cheese, sour cream, and/or diced avocado.