Preheat the oven to 350°F (177°C). Lightly grease a 9x13-inch baking dish using nonstick cooking spray.
In a large skillet over medium heat, warm the vegetable oil. Add the chopped onion and minced garlic, sautéing until the mixture becomes soft and translucent.
Stir in the ground cumin, dried oregano, and black pepper. Cook, stirring constantly, for about 1 minute to release the spices' aroma.
Add the chopped green chiles along with their liquid, cream of chicken soup, cream of mushroom soup, and chicken broth. Whisk thoroughly until the sauce is smooth and fully combined.
Spoon approximately 1/2 cup of the sauce into the prepared baking dish, spreading it evenly across the bottom.
Arrange one-third of the tortilla strips over the sauce. Spread one-third of the remaining sauce over the tortillas, followed by half of the shredded chicken and one-third of the cheese.
Repeat the layering process: add another third of the tortillas, another third of the sauce, the remaining chicken, and another third of the cheese.
Finish the casserole with the final layer of tortillas, the last portion of sauce, and the remaining cheese spread evenly across the top.
Place the baking dish in the preheated oven and bake, uncovered, for 35 to 45 minutes, or until the center is heated through and the sauce is bubbling around the edges.