Cook the chicken. Add the water, whole onion, head of garlic, 1 tablespoon salt, bay leaves, 1 tablespoon oregano, 1 tablespoon chicken bouillon, chicken breasts, and chicken thighs to a large stockpot. Bring to a boil over medium-high heat, then reduce to medium-low. Cover and gently simmer for about 45 minutes, or until the chicken is fully cooked and tender.
Shred the chicken and strain the broth. Remove the cooked chicken from the pot and set aside until cool enough to handle, then shred into bite-size pieces. Remove the onion and garlic and reserve them for the chile sauce. Discard the bay leaves. Pour the remaining broth through a fine-mesh strainer to remove any foam, small pieces of chicken, or impurities, then return the strained broth to the pot.
Soften the chiles. While the chicken cools, place the guajillo and árbol chiles in a separate pot and cover with water. Bring to a boil and cook for about 5 minutes, or until softened.
Make the red chile sauce. Transfer the softened chiles to a blender. Carefully remove the cooked onion and garlic from the chicken broth and add them to the blender. Add 1 teaspoon oregano, 1 teaspoon salt, and 1 cup of the chile cooking water. Blend until completely smooth.
Add the chile sauce and hominy. Pour the blended chile mixture into the chicken broth, straining it through a fine-mesh strainer if you prefer a smoother broth. Add the drained and rinsed hominy, remaining 1 teaspoon chicken bouillon, and menudo seasoning, if using.
Simmer. Bring the pozole to a gentle boil, then reduce the heat and simmer for about 30 minutes. After the first 20 minutes, add the shredded chicken back to the pot and cook for the remaining 10 minutes.
Season and serve. Taste the broth and add additional salt if needed. Ladle the pozole into bowls and top with cilantro, sliced radishes, diced onion, and plenty of fresh lime juice. Enjoy!