Grill chicken breast on outdoor grill and cool. Dice into 1-inch dice and set aside.
Saut onion and garlic in vegetable oil for 5 minutes. Do not drain. Add cumin and chili powder and continue to saut for an additional 2 minutes. Add tomatoes, chicken broth and undrained green chilies and simmer. Squeeze juice of 1/2 lime into the mixture and add oregano, parsley, tomato sauce, cilantro and cubed chicken. While soup simmers for at
least an hour, halve corn tortillas and then cut crosswise into 1/2-inch strips. Heat vegetable oil over moderate heat until hot, but not smoking. Cook tortilla strips in batches, stirring until golden and crisp, about 1 minute. Transfer strips to paper towels to drain.
Season with coarse ground pepper. To serve, ladle into bowls and top with shredded cheddar cheese and corn tortillas strips.