Preheat the oven to 400°F. Line a baking sheet with foil.
Cut the papery top off of the head of garlic to expose the cloves. Place the garlic, sweet potatoes, and poblano peppers on the prepared baking sheet and drizzle lightly with oil. Roast for 3o minutes, then let cool until you can easily handle the potatoes.
Meanwhile, heat the remaining oil in a large pot over medium-high heat. Remove the chorizo from the casing and add to the pot. Break apart the chorizo as you would ground beef as it cooks. Cook until browned all over, about 5 minutes.
Add the onion to the skillet and saute until translucent. Stir in the spices and cook for 1 minute. Pour in the chicken stock and bring to a boil.
Once the sweet potatoes and peppers are cool enough to handle, use your hands to peel the skins from the potatoes and poblanos. The skins should come off easily. Chop the sweet potato into bite-sized pieces. Stem and seed the poblano peppers.
In a blender or food processor, combine the roasted garlic (squeeze it out of the papery exterior), 1 of the roasted poblano peppers, 1 chipotle pepper, and the diced tomatoes. Process until well-combined. Pour the mixture into the pot of chili and bring back to a boil. Allow the liquid to reduce slightly, then stir in the black beans, hominy, tomato paste, and chopped sweet potato. Reduce to a simmer and cook for 30 minutes. Season with kosher salt and black pepper, to taste.
As the chili simmers, process the remaining poblano pepper, avocado, and lime juice until smooth.
Serve the chili warm with a dollop of the Poblano-Avocado Crema, crumbled goat cheese, chopped cilantro, and sliced radish on top.