Step 1

If using dried hominy, place it in a large pot, add enough water to cover the hominy by at least 4 inches, and then toss in the head of garlic. (Don’t add salt before or during cooking or the hominy will toughen.) Bring the water to a boil and then reduce the heat, cover partially, and gently simmer over medium-low heat until the hominy has “bloomed,” or opened, 4 to 4 1/2 hours, skimming the foam from the surface and adding more water as needed. The hominy will be chewy. Remove from the heat and add 2 teaspoons salt.If using precooked or canned hominy, dump the drained and rinsed hominy into a large pot and add 2 cups cold water.

Step 2

Place the chicken in a large pot and add enough water to cover by at least 2 inches. Add the onion, cilantro, and 1 tablespoon salt. Bring to a boil and then reduce the heat, cover partially, and gently simmer over medium-low heat until the chicken is cooked through and tender, about 40 minutes. Remove the chicken from the pot and let cool. Strain and reserve the cooking liquid.

Step 3

When the chicken is cool enough to handle, remove and discard the skin and bones and shred the meat.

Step 4

Return the shredded chicken and its cooking liquid to the pot along with the hominy and place over medium heat until warmed through, about 10 minutes. It should be soupy. Taste and add more salt if necessary. Remove the pot from the heat and set it aside while you make the chile purée. (You can cool the posole to room temperature, cover, and refrigerate for up to 3 days.)

Step 5

Place the ancho and guajillo chiles in a medium saucepan, add just enough water to cover, and bring to a boil over medium-high heat. Simmer until softened and rehydrated, about 10 minutes.

Step 6

Place the chiles and 3/4 cup of their soaking liquid in a blender or food processor along with the onion, garlic, cumin, cloves, and salt and purée until smooth. Pass the purée through a fine-mesh strainer into a bowl, pressing on the solids with the back of a wooden spoon to extract as much liquid as possible. Discard the solids.

Step 7

Heat the oil in a medium saucepan over medium heat until hot but not smoking. Add the chile purée and bring to a boil, then cover partially and simmer, stirring occasionally, until thickened, 6 to 8 minutes. Remove from the heat.

Step 8

Reheat the posole over medium-high heat until it comes to a gentle simmer. Stir in the chile purée and cook for 20 minutes. Taste and adjust the salt.

Step 9

Ladle the red posole into bowls and pass the limes, radishes, lettuce, onion, ground chile, dried oregano, tortilla chips or tostadas, and refried beans in bowls so each diner can fancy the posole up as they like. One topping that probably ought not be optional is the lime—go ahead and squeeze it liberally over the posole. One taste of how that bright acidity rounds things out and you'll understand.