Heat olive oil in a large saucepan over medium heat. When hot, add onion and cook until softened, about 2 minutes, then stir in garlic just until fragrant, about 30 seconds. Pour in the broth, tomatoes with their juices, and all seasonings - cumin, oregano, chipotle hot sauce and chili powder. Bring this mixture to a boil. Season the chicken breasts with salt and pepper, then add them to the pan, making sure they're coated in sauce and arranged in a single layer. Lower the heat, cover the pan, and simmer gently for 20-25 minutes, turning the chicken once halfway through, until it's tender and cooked through.
Transfer the tender chicken to a cutting board. Keep the pan on the heat, increase it to medium-high, and let the sauce boil uncovered. Cook, stirring now and then, until it reduces to a thick salsa consistency, about 8-12 minutes. Remove from heat.
Meanwhile, shred the chicken with two forks, then stir it back into the reduced sauce. Heat through over medium heat, stirring to coat every piece with sauce. Serve as desired.
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Notes
This recipe yields about 4 cups of shredded Chicken Tinga.
Note that chipotle chile powder is made from only smoked and dried jalapeño peppers, not standard chili powder which is a blend of spices. As written, this chicken is fairly mild. For medium spiciness add another 1/2 teaspoon of chipotle chile powder; for extra spice, add an additional 1 teaspoon or more.
Be sure to note the weight of the chicken breasts you buy as they vary a lot. As you would guess, the larger the chicken breasts, the longer they will take to be tender and shreddable.
To use shredded rotisserie chicken or Shredded Chicken for this recipe, follow the instructions for the sauce, but reduce the covered cooking time to 20 minutes. Then reduce the sauce uncovered until thickened to the consistency of a thick salsa. Stir in shredded chicken; reduce heat and simmer 5-10 minutes to reheat the chicken and allow it to soak up plenty of flavor.
Storage: Keep in an airtight container in the refrigerator for up to 3 days or freeze for up to 3 months
To reheat: Warm gently in a covered pan on the stovetop, adding a splash of water or broth if needed to maintain moisture