Place the dried chilies and smashed garlic into a small saucepan and add enough water to cover. Bring to a simmer over medium heat and cook until the chilies soften, around 5-6 minutes. Remove from heat and set aside to cool.
While the chilies are simmering, heat the olive oil in a large soup pot or Dutch oven over medium-high heat. Add in the onion and garlic and cook, stirring occasionally, until the onion is soft and translucent, around 4-5 minutes. Season with salt and black pepper, to taste.
Add the ground cumin, Mexican oregano, smoked paprika, and cinnamon. Stir to combine and cook for 1-2 minutes, or just until the spices become fragrant.
Add in the chicken and reduce heat to medium. Cook until lightly browned on all sides, around 2-3 minutes per side.
Add in the bay leaves, chicken stock, and hominy. Increase the heat to medium-high and bring the mixture to a rapid simmer.
Once simmering, transfer the ancho and guajillo chilies mixture to a blender or a food processor and blend until smooth. Pour the chili mixture into the pot and reduce heat to just below medium. Partially cover with a lid and cook for at least 30 minutes or up to 4 hours to give the flavors more time to develop.
After at least 30 minutes, remove from heat and carefully shred the chicken using two forks while still in the pot. Remove the bay leaves and discard.
Ladle the hot soup into individual serving bowls and top with sliced radishes, fresh cilantro, or other desired toppings. Serve immediately with fresh lime wedges for squeezing. Enjoy!