Diced avocado, crema, crumbled queso fresco, thinly sliced green cabbage, chopped white onion, sliced radishes, fried tortilla strips, dried chile de arbol and lime wedges, for serving
Position a rack in the center of the oven and preheat to 350 degrees F.
Arrange the ancho, guajillo, pasilla and morita chiles on a rimmed baking sheet in an even layer and toast until beginning to brown (being careful not to char) and very fragrant and smells like toasted nuts, about 5 minutes. Let cool on the baking sheet.
Increase the oven to 450 degrees F. Spread the hominy in an even layer on a foil-lined rimmed baking sheet and roast, tossing once, until lightly toasted, very fragrant and just beginning to brown on the edges of the baking sheet, 15 to 20 minutes.
Meanwhile, heat the oil in a large Dutch oven or large heavy pot over medium-high heat. Add the pork and cook, turning occasionally, until browned on the two largest sides, 10 to 12 minutes. Transfer to a large plate.
Add the garlic, onions and 6 1/2 teaspoons salt to the same pot. Reduce the heat to medium and cook, stirring occasionally, until the onions are tender and beginning to brown, 8 to 10 minutes. Add the toasted chiles, oregano, peppercorns, clove and 10 cups water to the pot and bring to a boil. Nestle the pork into the pot; the liquid should come halfway up the sides of the pork. Arrange the cilantro, mint and bay leaves around the pork. Cover, reduce to low and cook until the pork is very tender and shreds easily, 2 to 2 1/2 hours.
Remove and discard the cilantro, mint and bay leaves. Transfer the pork to a large plate. When cool enough to handle, shred the pork into large pieces. Loosely cover with plastic wrap.
Working in batches, puree the cooking broth and aromatics in a blender until completely smooth and no large pieces of chile remain. Season to taste.
Wipe the pot clean. Add the chile puree and toasted hominy and simmer over medium heat, uncovered, until the flavors meld, the broth is rich and the hominy has plumped up and is tender, about 20 minutes.
Stir the reserved pork into the pozole and cook until heated through, about 5 minutes.
Divide the posole among bowls. Top with desired toppings and a squeeze of lime.
The posole can be made up to 3 days ahead. Let cool; cover and chill.
When blending hot liquid, first let it cool for five minutes or so, then transfer it to a blender, filling only halfway. Put the lid on, leaving one corner open. Cover the lid with a kitchen towel to catch splatters, and pulse until smooth.