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Preparation
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Preheat and Prep: Preheat your oven to 375°F (190°C).
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Make the Roux: Add the remaining 2 tablespoons of butter to the skillet. Once melted, sprinkle in the flour and cook for 2 minutes, stirring constantly to form a paste.
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Create the Base Sauce: Gradually pour in the chicken broth while stirring continuously to avoid lumps. Bring the mixture to a gentle simmer until it thickens slightly.
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Puree the Sauce: Carefully puree the sauce until completely smooth using an immersion blender. Alternatively, transfer the mixture to a standard blender or food processor, blend until smooth, and return it to the skillet.
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Finish the Sauce: Remove the skillet from the heat. Stir in the sour cream, 1/2 cup of cilantro, and salt until fully incorporated and creamy. Set aside.
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Assemble the Dish: Spread about 1 cup of the prepared sauce in a thin, even layer on the bottom of a 9×13-inch baking dish.
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Fill the Tortillas: Working with one warmed tortilla at a time, place a portion of the shredded chicken and a pinch of cheese (reserving at least 1 cup of cheese for the top) down the center. Roll the tortilla tightly and place it seam-side down in the baking dish. Repeat with the remaining tortillas.
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Bake: Pour the remaining poblano sauce evenly over the assembled enchiladas. Sprinkle the reserved 1 cup of cheese over the top.
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Serve: Let cool for 5 minutes. Garnish with fresh cilantro and serve immediately.