Heat the oven to 375°F. Lightly grease a 9×13-inch baking dish. Wrap the tortillas in foil and warm them in the oven for 8 minutes, then keep wrapped so they stay pliable.
Char the poblanos: set them under a hot broiler or over an open flame, turning until blistered on all sides, 5–7 minutes. Transfer to a bowl, cover, steam 5 minutes, then peel, seed, and chop.
Build the saucy filling: In a large skillet, heat the oil over medium heat. Add the onion and 1/2 teaspoon of the salt; cook until translucent, 4–5 minutes. Stir in the garlic for 30 seconds. Sprinkle in the cumin and chili powder; toast 30 seconds. Measure out 1/2 cup of enchilada sauce and set aside for topping; pour the remaining 2 1/2 cups into the skillet along with the lime juice. Simmer 3 minutes.
Fold in the chicken, black beans, corn, green chiles, and chopped poblano. Simmer 2–3 minutes more to thicken slightly. Season with the remaining salt and the black pepper to taste, then remove from heat.
Assemble (three layers): Spread 2–3 tablespoons of the reserved sauce over the bottom of the baking dish. Arrange 3–4 warm tortillas in a snug layer, tearing to fit. Spoon on half of the filling and scatter with one-third of the cheeses. Add another tortilla layer, the remaining filling, and another third of the cheeses. Top with the final tortillas, brush/spread the remaining reserved sauce over them, and finish with the remaining cheese.
Cover the dish loosely with foil (do not press onto the cheese). Bake for 20 minutes. Uncover and bake 8–10 minutes more until bubbling and lightly browned. For extra color, broil 1–2 minutes, watching closely.
Rest 10 minutes before slicing so the layers set. Serve warm, topped with cilantro and a dollop of sour cream if you like.