Greek yogurt and cilantro, for topping, optional.
Put the chicken in a large saucepan and cover with 8 cups of cold water. Add the onion, 2 smashed cloves garlic and bay leaf and bring to a boil. Reduce the heat to a simmer and cook, stirring occasionally and skimming any foam that may appear on the surface, until the chicken is cooked through, about 30 minutes. Remove the chicken to a plate to cool for 25 minutes; reserve the broth.
Discard the skin and bones, shred the meat and reserve it for later. Strain and reserve the broth into a large measuring cup or bowl (you should have 6 cups, supplement with water if needed); discard the onion, garlic and bay leaf.
Heat a large saucepan over medium-high heat. Add the oil and when it is hot, add the sweet onion, 4 chopped cloves garlic and jalapeno and cook, stirring, until golden brown, about 5 minutes. Add the chili powder, cumin, cayenne pepper, and some salt and pepper and stir until fragrant, about 1 minute. Add the diced tomatoes, black beans and the reserved broth. Season the soup with a pinch of salt and pepper. Bring to a simmer and cook for 30 minutes. Stir in the chicken and corn tortillas and continue simmering until the chicken is heated through and the corn tortillas soften, about 15 minutes more.