To start:

Place olive oil in a large dutch oven over medium-high heat; when hot, add onion and sauté until softened, 8 to 10 minutes. Add garlic and cook until fragrant, 2 to 3 minutes.

Pork Version:

Sprinkle pork ribs with salt and add to pot; add additional chicken stock as needed to cover. Bring to a simmer and cook, skimming any foam that rises to the top, for 10 minutes; add the hominy with liquid, bay leaves, oregano, and toasted cumin, cover tightly, and cook, stirring occasionally, until pork shreds easily, about 90 minutes; remove ribs to a platter and keep stock simmering. When cool enough to handle, shred pork and return to pot, discarding fat and bones. Add additional chicken stock as needed to keep ingredients well covered. 

Meanwhile, cut stems from chiles and discard; cut open lengthwise and remove and discard seeds and membranes. Wash hands thoroughly after handling. Place chiles in a pot and cover with water; bring to a brisk simmer, cover, and cook for 30 minutes, stirring occasionally, until chiles are soft.

Transfer chiles to a blender, add some of the cooking water, and blend until finely ground; it should be the consistency of a thick shake. Add about a cup of the ground chile to pork and hominy; simmer for 10 minutes. Taste and adjust seasoning; depending on the salt content of the chicken broth and hominy, you may not need salt.

To serve, have guests add garnish as they desire to large soup bowls; ladle hominy, pork, and broth over the top. Pass additional garnish at the table.

Tip: when I'm making the chicken version, I often poach more chicken than needed for the posole; the chicken is lightly seasoned and will find its way into salads, sandwiches, or other soups.

Chicken version(which is much lighter):

Cook the onion and garlic as directed above; lightly salt the chicken and add to the onion mixture along with 8 cups stock. Set heat to medium, bring to a medium simmer, adjust heat to maintain, and cook chicken for 20 minutes. Remove chicken to a platter and set aside. Add the hominy with liquid, bay leaves, oregano, and toasted cumin, cover tightly, and cook, stirring occasionally, for about 45 minutes.

Prepare the chiles as directed above, adding about 1 cup of the purée to the pot. Simmer for 10 minutes; taste and adjust seasoning. You may or may not need salt.

When the chicken is cool enough to handle, slice it thinly across the grain. To serve, have guests add sliced chicken and garnishes as desired to wide soup bowls; ladle hot stock and hominy over the top. Pass additional garnish at the table.