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In a fry pan, cook the chicken and onion together in oil over medium-high heat until the chicken is just done.
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Divide the cooked chicken evenly between 8 tortillas; add 1 1/2 tablespoons cheese to each tortilla.
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Roll the enchiladas and place seam-side down in 9x13 baking dish that has been lightly sprayed with no-stick cooking spray.
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Melt the butter in a medium saucepan; stir in the flour to make a roux; stir and cook until bubbly; gradually whisk in the chicken broth and bring to boiling, stirring frequently.
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Remove from the heat; stir in the sour cream and green chiles; pour the sauce evenly over the enchiladas.
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Top with the remaining 3/4 cup cheese and bake at 400F for 20 minutes until the cheese is melted and the sauce near the edges of the baking dish is bubbly.