Soften the Chiles: Place the guajillo and ancho chiles in a small pot of water. Bring to a boil, then reduce the heat and simmer for about 10 minutes, or until softened.

Prepare the Chile Sauce: Let the chiles cool until safe to handle. Remove the stems and seeds, then blend the chiles with ½ cup of their cooking water until smooth.

Cook the Chicken: Place the chicken in a large stockpot with the halved onions, unpeeled garlic, bay leaves, and 1 cup of cilantro, including the stems. Add enough cool water to cover the ingredients. Cover, bring to a boil, then reduce to a simmer and cook for 35–40 minutes, or until the chicken is fully cooked.

Fry the Chile Puree: While the chicken cooks, heat a thin layer of olive oil in a large skillet over medium heat. Add the chile puree and cook for about 8 minutes, stirring occasionally, until it darkens and thickens.

Shred the Chicken: Remove the chicken from the stockpot and set aside to cool. Remove the onion, garlic, cilantro, and bay leaves from the broth. Once cool enough to handle, discard the chicken skin and bones, shred the meat, and return it to the pot.

Add the Hominy and Chile Sauce: Stir the chile puree, drained hominy, and salt into the broth.

Simmer the Pozole: Simmer uncovered for 30–45 minutes, stirring occasionally. The broth will develop a deeper flavor and become thicker as it cooks.

Prepare the Toppings: While the soup simmers, slice the radishes, chop the cilantro, dice the onions, and cut the limes into wedges.

Serve: Ladle the hot pozole into bowls and let everyone add their favorite toppings. Serve with tostadas on the side.