In a large saucepan or Dutch oven, heat the vegetable oil over medium-low heat. Add the minced onion, minced Anaheim pepper, and minced garlic and cook slowly, stirring occasionally, for 15 minutes, or until the minced onion begins to turn translucent.
Whisk the cornstarch into the chicken broth, then add the broth mixture to the pot, along with the cooked skinless, boneless chicken breast, cooked skinless, boneless chicken thigh, shredded mild cheddar cheese, minced corn tortillas, lime juice, diced tomatoes, ground cumin, ground cayenne pepper, Mexican oregano, and salt.
Increase the heat to medium and stir constantly to help the cheese melt. When the soup begins to bubble, reduce the heat to medium-low and simmer for 20 minutes.
Serve the soup topped with Tortilla strips and extra shredded cheese. Leftovers can be stored in the refrigerator, in a covered container, for up to 4 days.
