Preheat oven to 375°F (190°C).
Heat olive oil in a large skillet over medium heat. Add chopped onion and cook until softened, about 3-4 minutes.
Add minced garlic and cook for 30 seconds until fragrant.
Stir in shredded chicken, black beans, corn, salsa, ground cumin, chili powder, smoked paprika, salt, and black pepper. Cook for 5 minutes, stirring occasionally, until heated through and well combined.
In a 9×9-inch (23×23 cm) baking dish, spread a thin layer of the chicken mixture on the bottom.
Layer half of the tortilla strips evenly over the chicken mixture.
Sprinkle one cup of shredded cheddar cheese and half a cup of shredded Monterey Jack cheese over the tortillas.
Repeat layering with the remaining chicken mixture, tortilla strips, and cheeses.
Cover the casserole tightly with aluminum foil and refrigerate if making ahead. When ready to bake, remove from refrigerator and let sit at room temperature for 15 minutes.
Bake uncovered in the preheated oven for 25-30 minutes, or until the cheese is melted and bubbly and the casserole is heated through.
Remove from oven and let rest for 5 minutes before serving.
Garnish with chopped fresh cilantro and serve with sour cream and lime wedges on the side.