In a small dry skillet over medium heat, add the chiles. Cook, turning frequently, until fragrant and lightly blistered in places, 1-2 minutes total for smaller chiles and 4-5 minutes total for larger ones; remove. Discard their stems and shake out and discard the seeds.

In a small saucepan, bring 1 cup of water to a boil. Add the chiles and cook until softened, 3-4 minutes.

Transfer the chiles and the cooking liquid to a small food processor or blender; carefully purée until smooth, then set aside.

In a large, heavy pot, add the pork, 5 cups cold water, half of the onion, the garlic, salt, pepper, Mexican oregano, bay leaves, and smoked paprika. Bring to a boil over medium-high heat. Lower the heat to maintain a simmer and cook until the pork is very tender, about 3 hours. Remove from heat and use a strainer or a slotted spoon to transfer the pork to a large heatproof bowl (reserve the broth). When the pork is cool enough to handle, pull it into large shreds.

Set the strainer in the bowl, and strain the broth over the meat (discard the solids). Return the meat and broth to the pot and set it back over medium-high heat. Add the hominy and the prepared chile purée and bring to a simmer. Cook until the hominy is hot and the broth is thickened,15-20 minutes.

Finely dice the remaining onion half. Divide the posole between 8-10 wide soup bowls. Top with crema or sour cream, chopped onion, cilantro, avocado, cabbage, and a squeeze of lime if desired. Serve immediately.