Preheat the oven to 375°F (190°C).
Heat olive oil in a medium skillet over medium heat. Add the chopped onion and cook for 4-5 minutes until softened.
Add the minced garlic to the skillet and cook for 1 minute until fragrant.
Stir in the diced tomatoes with green chilies (including the juice), salsa verde, ground cumin, chili powder, smoked paprika, salt, and black pepper. Simmer for 5 minutes, stirring occasionally.
In a large mixing bowl, combine the cooked shredded chicken with half of the tomato sauce mixture.
Lightly grease a 9×13-inch baking dish. Spread a thin layer of the remaining tomato sauce on the bottom of the dish.
Layer half of the tortilla strips evenly over the sauce in the baking dish.
Spread half of the chicken and tomato sauce mixture over the tortilla layer.
Sprinkle with half of the shredded cheddar and Monterey Jack cheeses.
Repeat the layering with the remaining tortilla strips, chicken mixture, and cheeses.
Cover the casserole with aluminum foil and bake for 20 minutes.
Remove the foil and bake for an additional 15 minutes until the cheese is melted and bubbly and the tortillas are slightly crispy on top.
Remove from the oven and let the casserole rest for 5 minutes.
Sprinkle chopped fresh cilantro over the casserole before serving.