Spray a 9x13-inch baking dish with nonstick cooking spray.
In a mixing bowl, combine 1 cup of shredded Monterey Jack cheese with the cooked shredded chicken.
Lay out the flour tortillas and fill each with a generous portion of the chicken and cheese mixture. Roll them up tightly and place seam-side down in the baking dish.
In a skillet over medium heat, melt the butter. Stir in the flour and cook for about one minute to create a roux.
Gradually whisk in the chicken broth until smooth, then cook until slightly thickened.
Stir in the sour cream and diced green chiles. Season with salt and pepper to taste.
Pour the sauce over the enchiladas in the baking dish, ensuring they are well covered.
Sprinkle the remaining 1 cup of Monterey Jack cheese on top.
Bake for 20 to 25 minutes, or until the cheese is melted and bubbly.
Garnish with cilantro before serving.
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Notes
Store leftovers in an airtight container in the refrigerator for up to 3-4 days.