Preheat oven to 400 degrees F. Place tortillas directly on the oven rack in a single layer, baking until toasted and crisp, about 5 minutes; let cool 5-10 minutes before breaking into pieces.
Heat olive oil in a large cast iron skillet over medium heat. Add onion and poblano, and cook, stirring frequently, until tender, about 3-4 minutes. Stir in garlic and cumin until fragrant, about 1 minute.
Stir in diced tomatoes, enchilada sauce and chicken; season with salt and pepper, to taste.
Bring to a boil; reduce heat and simmer, stirring occasionally, until flavors have blended, about 5 minutes.
Stir in tortillas until evenly combined; sprinkle with cheeses.
Place into oven and bake until bubbly and cheeses have melted, about 15-20 minutes.