To a medium pot, add the chicken broth and bring to a simmer over medium heat. Add the chicken breasts and turn the heat down to medium-low (or until the broth is barely simmering) and cook until the chicken is cooked, tender, and reaches an internal temperature of 165 F on an instant read meat thermometer.
Use tongs to move the chicken to a plate, and let it cool before shredding it. Reserve the cooking liquid.
Add the tomatillos, onion, poblano peppers, jalapeños, cilantro, garlic, and oregano to a food processor. Blend until the mixture is smooth.
Heat a large lidded pot over medium heat, then add the olive oil and swirl to coat the bottom of the pot. Add the tomatillo mixture and salt, and mix. Cover the pot (it will splatter while it cooks) and saute until the peppers and onions are cooked. You can tell the peppers and onions are cooked because they won’t have a pungent raw flavor anymore and the color will change from a vibrant green to a yellowish green. This will take about 10-15 minutes.
Add the reserved cooking liquid, and stir to combine. Stir in the shredded chicken and drained hominy, and simmer until the hominy is heated through (about 15 minutes).