In a large saucepan or Dutch oven, heat the vegetable oil over medium-low heat. Add the minced onion, pepper, and 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 chicken, cheddar cheese, minced tortillas, lime juice, diced tomatoes, cumin, cayenne pepper, 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.
To freeze this soup, allow it to cool completely, then place in a freezer-safe container and freeze for up to 3 months. Thaw overnight in the refrigerator then reheat over low heat on the stove, or in the microwave.
Poblano pepper can be substituted for Anaheim.
If you cannot find Mexican oregano, you can substitute regular.
You can use half cheddar cheese and half Monterey Jack for a varied flavor.
This soup is an excellent candidate for the slow cooker! Complete step 1, then transfer all of the ingredients to a slow cooker and cook on low for 4 to 6 hours.