Heat the broiler and place tomatillos, onion, jalapeno and garlic on a rimmed baking sheet. Broil for 8 to 10 minutes, turning once, until softened with dark patches; watch the garlic and remove it early if browned.
Blend the roasted vegetables with cilantro and 1 cup broth until mostly smooth. Heat oil in a large soup pot and cook the green mixture for 3 minutes, stirring.
Add remaining broth, chicken, cumin and salt. Simmer gently for 20 to 25 minutes until the chicken reaches 165°F, then remove the meat to shred.
Add hominy and simmer for 10 minutes. Return the shredded chicken and stir in lime juice.
Divide among bowls and add the measured cabbage and radish slices evenly just before serving.