Preheat the oven to 275˚F.
Rub the pork all over with chilies, salt, pepper, and minced garlic. Dot with the cloves. Set in a large roasting pan, and cover with aluminum foil. Cook about 4 hours, until the meat is falling off the bone. Remove from the oven and let sit until cool enough to handle. Pull the meat off the bone and shred.
While the meat is cooking, combine the sliced radishes, a pinch of salt, and juice from 3 of the limes in a medium bowl. Set aside. This is best done about an hour before serving. Place 3/4 of the diced red onions in a large bowl, and fold in 4 cups of the chopped cilantro. Set aside to use as a garnish.
Turn on the broiler. Set the tomatillos on the rack of a broiler pan 1 to 2 inches from heat. Broil, turning once, until tomatillos are softened and slightly charred, about 7 minutes. Place in the bowl of a food processor. Add the juice from the remaining 3 limes, the last diced red onion, the last 2 cups of cilantro, and the avocado to the food processor. Pulse until puréed. Taste and add more salt or lime juice as needed.
Serve the carnitas with warmed tortillas, onion and cilantro garnish, salted radishes, tomatillo salsa, queso fresco, and lime wedges.