Place the chicken, roughly cut onion, pressed garlic, and 1 tsp salt in a large skillet. Add enough water to just cover the chicken.

Bring to a boil and then lower to a simmer. Cover the pan and simmer for 20 to 25 minutes. Use a slotted spoon to remove the chicken to a platter. Let rest until cool enough to handle. Use your fingers, or two forks, to remove the skin and then pull the meat from the bones and roughly shred it. Set aside.

Carefully strain the broth through a colander into a heat-proof bowl and remove 1 cup of the broth to use for cooking the tinga chicken. Save the remaining broth for another use. (It freezes perfectly!).

In a large skillet (can be the same one use poached the chicken in), heat the oil over medium heat. Add the onions and cook until soft, about 4 to 5 minutes. Add the garlic and cook, stirring often, for another 30 seconds.

Stir in the oregano and cumin. Cook, stirring, for about 1 minute.

Add the tomatoes (with juice), chicken broth, peppers, and adobo sauce. Stir in the salt and add the bay leaves. Let simmer for about 15 minutes.

Remove the bay leaves, and then carefully transfer the tinga sauce to a blender (a ladle helps). Purée for about 1 minute until smooth. Add the puréed sauce back into the skillet. Stir in the chicken and cook, stirring often, until fully heated through, about 5 minutes.

Serve at once with warmed tortillas (see NOTES), cheese, cilantro, red onions, and hot sauce, if desired.