Cook the chicken: Place chicken, half the onion, 3 garlic cloves, bay leaves, and a pinch of salt in a large pot. Add broth to cover. Bring to a boil, then reduce to a simmer. Cover and cook 25-30 minutes until chicken is tender. Remove chicken, reserve broth, and shred chicken once cool enough to handle.
Make the salsa verde: While chicken simmers, cook tomatillos, poblanos, and jalapeño in a skillet with a splash of water over medium heat for 8-10 minutes until softened. Transfer to a blender with remaining onion, remaining garlic, cilantro, oregano, cumin, and 1 cup of reserved broth. Blend until completely smooth.
Cook the salsa base: Heat avocado oil in the same skillet over medium heat. Carefully pour in the blended salsa verde (it may splatter). Cook, stirring often, for 8-10 minutes until slightly darkened and thickened.
Combine the soup: Pour the cooked salsa verde into the pot with the reserved chicken broth. Stir well and bring to a gentle simmer. Add drained hominy and shredded chicken. Simmer, uncovered, for 15-20 minutes to let flavors meld. Season with salt and pepper to taste.
Garnish and serve: Ladle hot pozole into bowls. Top with shredded cabbage, sliced radishes, diced avocado, and a squeeze of lime. Add tortilla strips, extra onion, cilantro, or cotija cheese if desired. Serve immediately with toppings on the side.