Add the oil, garlic, and tortillas to a small stock pot or dutch oven set over a medium heat. Sauté until the tortillas just start to crisp up and the garlic starts to brown slightly, about 4 to 5 minutes. Add ancho chilies, jalapeño, cumin, coriander, cilantro, and bay leaf; cook for a few minutes. Add onion puree, tomato puree, chicken stock, and cracked black pepper. Bring mixture to a boil and then bring the soup to a simmer and let simmer for 40 to 45 minutes. Skim fat from the surface of the soup as necessary.
Garnish each soup bowl with smoked chicken, green cabbage, fried tortilla strips, jalapeño pepper, red radish, avocado, and white cheddar.
After soup has simmered, finish flavoring soup with cayenne pepper, lemon juice, and salt. Process mixture in a food mill or blender until smooth. If the soup is too thick, thin with a little bit of chicken stock.
Pour hot soup over garnish, and serve immediately.