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Preheat oven to 375° and have ready a 9x13” greased baking dish. Place the halved tortillas in a shallow dish (like a pie dish), and pour over 1 cup of chicken broth. Set aside and allow to soak.
In a large rimmed sauté pan over medium-high heat, melt together butter and olive oil. Add celery, onion, and bell pepper and sauté until softened and onions are translucent, 5-8 minutes.Add mushrooms and sauté until juices have released and they begin to take on a brown color, about 10 minutes. Add the garlic and sauté 30 seconds to a minute.
Sprinkle over the flour and stir until a paste coats the vegetables. Slowly drizzle in the remaining 4 cups of chicken broth, whisking constantly until flour has completely dissolved and no lumps remain. Allow mixture to come to a boil, then reduce the heat to medium and allow to simmer about 5 minutes, or until the mixture has thickened.
Stir in the tomatoes and green chilis and allow to simmer for another five minutes. Add the chili powder, cumin, salt, pepper and 2 tablespoons of cilantro. Stir once more, remove from heat and set aside.
Combine the Cheddar and Monterrey Jack in a small bowl, and set aside.
Spoon a thin layer of sauce on the bottom of the dish and then add a layer of tortillas. Add a layer of sauce over the tortillas, followed by a layer of chicken and then a third of the cheese mixture. Repeat with another layer of tortillas, sauce, chicken, and half of the remaining cheese.
Add one more layer of tortillas, along with the remaining sauce, and cheese. Bake for 25-30 minutes until bubbly. Sprinkle with the remaining cilantro and set aside to cool for at least 15 minutes.