Preheat oven to 350°F. Lightly grease a 9x13-inch baking dish.
Melt butter in a large skillet over medium heat. Add onion, poblano pepper, and bell pepper and cook until softened, about 5 minutes. Add garlic and cook for 30 seconds.
Stir in Rotel, cream of chicken soup, cream of mushroom soup, chicken broth, cumin, chili powder, smoked paprika, salt, and pepper. Cook until smooth and heated through, about 3 to 4 minutes.
Remove from heat. Stir in the sour cream until smooth, then fold in the shredded chicken.
Spread one-third of the tortilla pieces in the bottom of the prepared baking dish.
Top with one-third of the chicken mixture and one-third of the cheese.
Repeat the layers two more times, ending with the remaining cheese on top.
Cover loosely with foil and bake for 25 minutes.
Remove foil and bake for another 15 minutes, or until hot, bubbly, and lightly browned on top.
Let rest for 10 minutes before serving. Garnish with cilantro if desired.