In a medium-sized saucepan, warm the olive oil over medium heat. Add the chopped onion and sauté until softened and fragrant, about 3 to 4 minutes. Stir in the flour and cook for 1 minute, stirring constantly to form a light roux.

Stir in the diced green chilies, shredded chicken, chicken broth, and garlic powder. Bring the mixture to a gentle simmer and let it cook uncovered for 5 minutes, allowing the flavors to meld together.

In a small bowl, dissolve the cornstarch in 1/4 cup of cold water, ensuring there are no lumps. Pour this slurry into the soup and cook for an additional 2 minutes, stirring continuously, until the soup thickens slightly.

Tear the corn tortillas into strips, then cut each strip into thirds. Add the tortilla pieces to the soup and stir gently. Let them heat through for a few minutes, stirring occasionally so they soften evenly without sticking together.

Sprinkle the shredded cheddar cheese across the surface of the soup. Cover the saucepan with a lid and let it sit over low heat until the cheese is fully melted and gooey. Serve the soup piping hot, garnished as desired.