Place chicken, broth, onion halves, garlic, bay leaves, oregano, and salt in a large Dutch oven. Bring to a boil, reduce to a simmer, and cook for 35-40 minutes until chicken is tender. Remove chicken and strain broth, discarding solids.
Broil poblanos and jalapeños on a baking sheet for 8-10 minutes, turning halfway, until charred. Transfer to a bag to steam for 10 minutes, then peel, remove stems and seeds.
Boil tomatillos in water for 8-10 minutes until soft and olive-colored. Drain and cool slightly. Blend with roasted peppers, onion, garlic, and cilantro until completely smooth.
Heat oil in the same pot over medium-high heat. Carefully pour in the green sauce and cook, stirring constantly, for 5-7 minutes until darkened and fragrant. Stir in cumin and season with salt and pepper.
Pour the reserved broth back into the pot. Add the rinsed hominy and bring to a gentle boil. Reduce heat and simmer uncovered for 20 minutes.
Shred the cooked chicken with two forks, discarding skin and bones. Add the shredded chicken to the pot and simmer for 5 more minutes. Adjust seasoning to taste.
Ladle the pozole into deep bowls and serve with toppings like cabbage, radishes, avocado, lime wedges, oregano, onion, tostadas, and sour cream on the side for customizing.