Begin by cooking the chicken. Place the chicken thighs in a large stockpot and pour in the chicken broth until fully submerged. Add the bay leaves, quartered onion, garlic cloves, salt, and black pepper. Bring the mixture to a boil, then reduce to a gentle simmer. Partially cover and cook until the chicken is tender and cooked through, about 30 minutes. Remove the onion, garlic, and bay leaves from the pot using a slotted spoon, leaving the flavorful broth behind. Transfer the chicken to a plate to cool slightly, then shred the meat, discarding the skin and bones.
Next, roast the peppers to develop their smoky flavor. Preheat your broiler and lightly coat a baking sheet with cooking spray. Slice the poblano, serrano, and optional jalapeño peppers in half lengthwise. Remove stems, seeds, and veins — or leave some intact for a spicier broth. Arrange them cut side down on the baking sheet and broil for 7–10 minutes, until the skins are blistered and charred. Immediately transfer the roasted peppers to a plastic bag, seal it, and allow them to steam for 5–10 minutes. Once cooled slightly, peel off the outer skins — they should come off easily.
Prepare the green sauce by blending the roasted peppers, tomatillos, chopped onion, fresh cilantro, and three garlic cloves. Add about ½ cup of the reserved chicken broth to the blender and process until smooth and velvety. Pour this vibrant green mixture into the pot of remaining broth.
Stir in the shredded chicken, hominy, dried oregano, ground cumin, and additional salt and pepper to taste. Simmer for another 5–10 minutes, allowing the flavors to meld and the soup to thicken slightly.
To serve, ladle the hot pozole into bowls. Garnish each serving with a handful of shredded cabbage, radish slices, diced onion, creamy avocado, a dollop of sour cream, and a squeeze of fresh lime. Finish with a handful of tortilla strips or crushed tortilla chips for added crunch. Serve warm and enjoy the bold, comforting layers of this traditional Mexican favorite.