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In a large skillet set over medium heat, add 1 tablespoon of butter, the diced onion & red pepper, and minced garlic. Cook for 2-3 minutes then remove the mixture from the skillet and place in a bowl. Set the bowl aside.
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In the same skillet set over medium-low heat, melt the 3 remaining tablespoons of butter then add flour and whisk together. Be careful not to let the roux brown too much, you are looking for it to turn a golden brown color.
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Let the roux cook for 5 minutes then whisk in a cup of the hot chicken broth ensuring that there are no lumps. Then whisk in the remaining chicken broth and the sour cream.
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Add 1 diced poblano pepper to the cream mixture and then purée it using an immersion (stick) blender or an upright blender.
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Add in all of the seasonings, the black beans, and the onion, red bell pepper, and garlic mixture. Simmer on medium-low heat for 5 minutes, then add the shredded chicken, the other diced roasted poblano pepper, diced tortillas and half of the cheeses. Stir it until it is fully combined.
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Turn on your oven’s broiler to high. Top the enchilada skillet with the remaining cheese and stick the skillet under the broiler for about 3-5 minutes, or until the cheese is golden brown.
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Serve the skillet with optional toppings if you desire.